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JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

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Q.1. Let A(α,−2),B(α, 6) and C(α/4, −2) be vertices of a ΔABC. If (5, α/4) is the circumcentre of ΔABC, then which of the following is NOT correct about ΔABC ?        (JEE Main 2022)
(a) area is 24
(b) perimeter is 25
(c) circumradius is 5
(d) inradius is 2

Ans. b
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Circumcentre of ΔABC
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Circumradius = 10/2 = 5
r = Δ/s = 24/12 = 2


Q.2. Let m1, m2 be the slopes of two adjacent sides of a square of side a such that a2 + 11a + 3(m1+ m22) = 220. If one vertex of the square is (10(cos⁡α − sin⁡α), 10(sin⁡α + cos⁡ α)), where α ∈(0, π/2) and the equation of one diagonal is (cos⁡α − sin⁡α)x + (sin⁡α + cos⁡α)y = 10, then 72(sin4⁡α + cos4⁡α) + a2 − 3a + 13 is equal to:    (JEE Main 2022)
(a) 119
(b) 128
(c) 145
(d) 155

Ans. b


Q.3. Let the circumcentre of a triangle with vertices A(a, 3), B(b, 5) and C(a, b), ab > 0 be P(1,1). If the line AP intersects the line BC at the point Q(k1, k2), then k+ k2 is equal to:    (JEE Main 2022)
(a) 2
(b) 4/7
(c) 2/7
(d) 4

Ans. b
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let D be mid-point of AC, then
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let E be mid-point of BC,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
On putting b=−1, we get a = 5 or −3
But a = 5 is rejected as ab > 0
A(−3,3), B(−1,5),C(−3, −1),P(1, 1)
Line BC ⇒ y = 3x + 8
Line AP ⇒ y = 3 − x/2
Point of intersection
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.4. The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 39 and x − y = 3 respectively and P(2, 3) is its circumcentre. Then which of the following is NOT true?    (JEE Main 2022)
(a) (AC)2 = 9p
(b) (AC)2 + p2 = 136
(c) 32 < area(ΔABC) < 36
(d) 34 < area(ΔABC) < 38

Ans. d
Intersection of 2x + y = 0 and x - y  = 3 : A(1, - 2)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Equation of perpendicular bisector of AB is
x − 2y = −4
Equation of perpendicular bisector of AC is
x + y = 5
Point B is the image of A in line x − 2y + 4 = 0 which is obtained as B(−13/5, 26/5)
Similarly vertex C : (7,4)
Equation of line BC : x + 8y = 39
So, p = 8
AC = (7−1)2+(4+2)2=62
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Area of triangle ABC = 32.4


Q.5. Let A(1,1),B(−4,3),C(−2,−5) be vertices of a triangle ABC, P be a point on side BC, and Δ1 and Δ2 be the areas of triangles APB and ABC, respectively. If Δ1 : Δ2 = 4:7, then the area enclosed by the lines AP,AC and the x-axis is    (JEE Main 2022)
(a) 1/4
(b) 3/4
(c) 1/2
(d) 1

Ans. c
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
P (-20/7 , -11/7)
Line AC : y − 1 = 2(x − 1)
Intersection with x-axis =(1/2, 0)
Line AP : y − 1 = 2/3 (x−1)
Intersection with x-axis (−1/2, 0)
Vertices are (1, 1),(1/2, 0) and (−1/2, 0)
Area =1/2 sq. unit


Q.6. A point P moves so that the sum of squares of its distances from the points (1,2) and (−2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A, B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to:        (JEE Main 2022)
(a)9/2
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) 9

Ans. b
Let point P:(h,k)
(h − 1)2 + (k − 2)+ (h + 2)2 + (k − 1)2 = 14
2h2 + 2k2 + 2h − 6k − 4 = 0
Locus of P : x2 + y2 + x − 3y − 2 = 0
Intersection with x-axis,
x2 + x − 2 = 0
⇒ x = −2, 1
Intersection with y-axis,
y2 − 3y − 2 = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Area of the quadrilateral
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.7. Let the point P(α, β) be at a unit distance from each of the two lines L1 : 3x − 4y + 12 = 0, and L2 : 8x + 6y + 11 = 0. If P lies below L1 and above L2, then 100(α + β) is equal to        (JEE Main 2022)
(a) −14
(b) 42
(c) −22
(d) 14

Ans. d
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
L1 : 3x − 4y + 12 = 0
L2 : 8x + 6y + 11 = 0
Equation of angle bisector of L1 and L2 of angle containing origin
2(3x − 4y + 12) = 8x + 6y + 11
2x + 14y − 13 = 0 ...... (i)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ 3α − 4β + 7 = 0 ...... (ii)
Solution of 2x + 14y − 13 = 0 and 3x − 4y + 7 = 0 gives the required point P(α,β),α = −23/25, β = 53/50
100(α + β) = 14


Q.8. A line, with the slope greater than one, passes through the point A(4, 3) and intersects the line x − y − 2 = 0  at the point B. If the length of the line segment AB is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE, then B also lies on the line :        (JEE Main 2022)
(a) 2x + y = 9
(b) 3x − 2y = 7
(c) x + 2y = 6
(d) 2x − 3y = 3

Ans. c


Q.9. Let α1, α21 < α2) be the values of α fo the points (α, −3), (2, 0) and (1, α) to be collinear. Then the equation of the line, passing through (α1, α2) and making an angle of π/3 with the positive direction of the x-axis, is :        (JEE Main 2022)
(a) x − √3y − 3√3 + 1 = 0
(b) √3x − y + √3 + 3 = 0
(c) x − √3y + 3√3 + 1 = 0
(d) √3x − y + √3 − 3 = 0

Ans. b

Points A(α, −3), B(2, 0) and C(1, α) are collinear.
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∴ Slope of AB = Slope of BC
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ −3 = α(2 − α)
⇒ −3 = 2α − α2
⇒ α2 − 2α − 3 = 0
⇒ α− 3α + α − 3 = 0
⇒ α(α − 3) + 1(α − 3) = 0
⇒ (α + 1)(α − 3) = 0
⇒ α = −1, 3
Given, α1< α2
∴ α1 = 1 and α2 = 3
∴ (α1, α2) = (−1, 3)
Now, equation of the line passing through (−1, 3) and making angle π/3 with positive x-axis is
(y − y1) = m(x − x1)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.10. Let the area of the triangle with vertices A(1, α), B(α, 0) and C(0, α) be 4 sq. units. If the points (α, −α), (−α, α) and (α2, β) are collinear, then β is equal to        (JEE Main 2022)
(a) 64
(b) −8
(c) −64
(d) 512

Ans. c

∵ A(1, α), B(α, 0) and C(0, α) are the vertices of ΔABC and area of ΔABC = 4

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

⇒ |1(1 − α) − α(α) + α2| = 8
⇒ α = ±8
Now, (α,−α),(−α,α) and (α2, β) are collinear
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ 8(8 − β) + 8(−8 − 64) + 1(−8β − 8 × 64) = 0
⇒ 8 − β − 72 − β − 64 = 0
⇒ β = - 64


Q.11. Let R be the point (3, 7) and let P and Q be two points on the line x + y = 5 such that PQR is an equilateral triangle. Then the area of ΔPQR is:        (JEE Main 2022)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. d
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let, side of triangle = a.
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
From figure, h = a sin ⁡60   
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.12. If two straight lines whose direction cosines are given by the relations l + m − n = 0, 3l2 + m2 + cnl = 0 are parallel, then the positive value of c is :        (JEE Main 2022)
(a) 6
(b) 4
(c) 3
(d) 2

Ans. a
l + m − n = 0 ⇒ n = l + m
3l2 + m2 + cnl = 0
3l2 + m2 + cl(l + m) = 0
= (3 + c)l2 + clm + m2 = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∴ Lines are parallel
D = 0
c2 − 4(3 + c) = 0
c2 − 4c − 12 = 0
(c − 4)(c + 3) = 0
c = 4 (as c > 0)


Q.13. In an isosceles triangle ABC, the vertex A is (6, 1) and the equation of the base BC is 2x + y = 4. Let the point B lie on the line x + 3y = 7. If (α, β) is the centroid of ΔABC, then 15(α + β) is equal to:        (JEE Main 2022)
(a) 39
(b) 41
(c) 51
(d) 63

Ans. c
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
B(1, 2)
Let C(k, 4 − 2k)
Now AB2 = AC2
5+ (−1)= (6 − k)+ (−3 + 2k)2
⇒ 5k−2 4k + 19 = 0
(5k − 19)(k − 1) = 0 ⇒ k = 19/5
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Centroid (α, β)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Now 15(α + β)
15(17/5) = 51


Q.14. Let a triangle be bounded by the lines L1 : 2x + 5y = 10; L2 : −4x + 3y = 12 and the line L3, which passes through the point P(2, 3), intersects L2 at A and L1 at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to :        (JEE Main 2022)
(a) 110/13
(b) 132/13
(c) 142/13
(d) 151/13

Ans. b


Q.15. The distance of the origin from the centroid of the triangle whose two sides have the equations x − 2y + 1 = 0 and 2x − y − 1 = 0 and whose orthocenter is (7/3, 7/3) is :        (JEE Main 2022)
(a) 2
(b) 2
(c) 22
(d) 4

Ans. c
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
For point A,
2x − y − 1 = 0 ...... (1)
x − 2y + 1 = 0 ...... (2)
Solving (1) and (2), we get
x = 1, y = 1.
∴ Point A = (1, 1)
Altitude from B to line AC is perpendicular to line AC.
∴ Equator of altitude BH is
2x + y + λ = 0 ...... (3)
It passes through point H(7/3, 7/3) so it satisfy the equation (3).
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ α = −7
∴ Altitude BH = 2x + y − 7 = 0 ...... (4)

Solving equation (1) and (4), we get
x = 2, y = 3.
∴ Point B = (2, 3)
Altitude from C to line AB is perpendicular to line AB.
∴ Equation of altitude CH is
x + 2y + λ = 0 ...... (5)
It passes through point H(7/3, 7/3) so it satisfy equation (5).
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ λ = −7
∴ Altitude CH = x + 2y − 7 = 0 ...... (6)
Solving equation (2) and (6), we get
x = 3, y = 2
∴ Point C = (3, 2)
Centroid G (x, y) of triangle A (1, 1), B (2, 3) and C (3, 2) is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Now d Distance of point G (2, 2) from center O (0, 0) is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.16. The distance between the two points A and A' which lie on y = 2 such that both the line segments AB and A' B (where B is the point (2, 3)) subtend angle π/4 at the origin, is equal to:        (JEE Main 2022)
(a) 10
(b) 48/5
(c) 52/5
(d) 3

Ans. c


Q.17. The equations of the sides AB, BC and CA of a triangle ABC are 2x + y = 0, x + py = 15a and x − y = 3 respectively. If its ortho centre is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE, then p is equal to ______________.        (JEE Main 2022)

Ans. 3
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

∴ p2 − 8p + 15 = 0
∴ p = 3 or 5
But if p=5 then a=3 not acceptable
∴ p = 3


Q.18. Let JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE be a fixed point in the xy-plane. The image of A in y-axis be B and the image of B in x-axis be C. If D(3 cos⁡θ, a sin⁡θ) is a point in the fourth quadrant such that the maximum area of ΔACD is 12 square units, then a is equal to ____________.      (JEE Main 2022)

Ans. 8

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.19. A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of 7α + 3β is equal to ____________.        (JEE Main 2022)

Ans. 31
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Bisector of angle PAQ is X = 23/7

⇒ M = (23/7, 8/3)
So, 7α + 3β = 31


Q.20. A man starts walking from the point P(−3, 4), touches the x-axis at R, and then turns to reach at the point Q(0, 2). The man is walking at a constant speed. If the man reaches the point Q in the minimum time, then 50((PR)2 + (RQ)2) is equal to ____________.        (JEE Main 2021)

Ans. 1250
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
50(PR2 + RQ2)
= 50(20 + 5)
= 50(25)
= 1250


Q.21. Let the points of intersections of the lines x − y + 1 = 0, x − 2y + 3 = 0 and 2x − 5y + 11 = 0 are the mid points of the sides of a triangle ΔABC. Then, the area of the ΔABC is _________.        (JEE Main 2021)

Ans. 6
Intersection point of given lines are (1, 2), (7, 5), (2, 3)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.22. Consider a triangle having vertices A(−2, 3), B(1, 9) and C(3, 8). If a line L passing through the circum-centre of triangle ABC, bisects line BC, and intersects y-axis at point (0, α/2), then the value of real number α is __________.        (JEE Main 2021)

Ans. 9

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∠B = 90
Circum-center =(1/2, 11/2)
Mid point of BC =(2, 17/2)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Passing through (0, α/2)
α/2 = 9/2 ⇒ α = 9


Q.23. A square ABCD has all its vertices on the curve x2y2 = 1. The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is        (JEE Main 2021)

Ans. 80
x2y2 = 1
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Graph of this equation,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ p2q= 1
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE midpoint of AB lies
On x2y2 = 1
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ (p + q)2(p − q)2 = 16
⇒ (p− q2)2 = 16
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ p± 4p− 1 = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.24. Let tanα, tanβ and JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE be the slopes of three line segments OA, OB and OC, respectively, where O is origin. If circumcentre of ΔABC coincides with origin and its orthocentre lies on y-axis, then the value of JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE is equal to ____________.        (JEE Main 2021)

Ans. 144
Since orthocentre and circumcentre both lies on y-axis.
⇒ Centroid also lies on y-axis.

⇒ ∑ cos ⁡α = 0
cos α + cos β + cos γ = 0
⇒ cos3 α + cos3 β + cos3 γ = 3 cos α cos β cos γ
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.25. The maximum value of z in the following equation z = 6xy + y2, where 3x + 4y ≤ 100 and 4x + 3y ≤ 75 for x ≥ 0 and y ≥ 0 is __________.        (JEE Main 2021)

Ans. 904
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
3x + 4y ≤ 100
4x + 3y ≤ 75
x ≥ 0, y ≥ 0
Feasible region is shown in the graph
Let maximum value of 6xy + y2 = c
For a solution with feasible region,
6xy + y2 = c and 4x + 3y = 75 must have at least one positive solution.
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.26. Let λ be an integer. If the shortest distance between the lines x − λ = 2y − 1 = −2z and x = y + 2λ = z − λ is √7/2√2, then the value of | λ | is _________.         (JEE Main 2021)

Ans. 1
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Point on line = (0, −2λ, λ)
Distance between skew lines
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
= |10λ + 3| = 7
⇒ λ = −1


Q.27. Consider the lines L1 and L2 defined by L1 : x√2 + y − 1 = 0 and L2 : x√2 − y + 1 = 0
For a fixed constant λ, let C be the locus of a point P such that the product of the distance of P from L1 and the distance of P from L2 is λ2. The line y = 2x + 1 meets C at two points R and S, where the distance between R and S is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.  
The value of D is __________.     (JEE Advanced 2021)

Ans. 77.14
According to the question,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let R ≡ (x1, y1) and S(x2, y2)
∵ C cuts y − 1 = 2x at R and S.
So, |2x− 4x2| = 3λ2
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∴ |x− x2| = √6|λ|
and |y− y2| = 2|x− x2| = 2√6|λ|
∵ RS2 = 270 (given)
⇒ (x1 − x2)+ (y1 − y2)= 270
⇒ (√6λ)2 + (2√6|λ|)2 = 270
⇒ 30λ= 270 ⇒ λ2 = 9
Now, mid-point of RS is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE and slope of RS = 2 and slope of R′S′ = −1/2
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
On solving x + 2y − 2 = 0 with C, we get
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, D ≡ (R′S′)= (x1 − x2)+ (y1 − y2)2  

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.28. Consider the lines L1 and L2 defined by
L1 : x√2 + y − 1 = 0 and L2 : x√2 − y + 1 = 0
For a fixed constant λ, let C be the locus of a point P such that the product of the distance of P from L1 and the distance of P from L2 is λ2. The line y = 2x + 1 meets C at two points R and S, where the distance between R and S isJEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE Let the perpendicular bisector of RS meet C at two distinct points R' and S'. Let D be the square of the distance between R' and S'.
The value of λ2 is __________.       (JEE Advanced 2021)

Ans. 9
According to the question,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let R ≡ (x1, y1) and S(x2, y2)
∵ C cuts y − 1 = 2x at R and S.
So, |2x2 − 4x2| = 3λ2 
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∵ RS2 = 270 (given)
⇒(x1 − x2)+ (y1 − y2)2 = 270
⇒(√6λ)+ (2√6|λ|)2 = 270
⇒ 30λ2 = 270 ⇒ λ2 = 9


Q.29. Let A be the set of all points (α, β) such that the area of triangle formed by the points (5, 6), (3, 2) and (α, β) is 12 square units. Then the least possible length of a line segment joining the origin to a point in A, is:         (JEE Main 2021)
(a) 4/√5
(b) 16/√5
(c) 8/√5
(d) 12/√5

Ans. c
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
4α − 2β = ± 24 + 8
⇒ 4α − 2β = + 24 + 8 ⇒ 2α − β = 16
2x − y − 16 = 0 ..... (1)
⇒ 4α − 2β = − 24 + 8 ⇒ 2α − β = −8
2x − y + 8 = 0 ...... (2)
perpendicular distance of (1) from (0, 0)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
perpendicular distance of (2) from (0, 0)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.30. If p and q are the lengths of the perpendiculars from the origin on the lines, x cosec α − y sec α = k cot 2α and x sinα + y cosα = k sin 2α respectively, then k2 is equal to :        (JEE Main 2021)
(a) 4p2 + q2
(b) 2p2 + q2
(c) p2 + 2q2
(d) p2 + 4q

Ans. a
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ x cos ⁡α − y sin ⁡α = k/2 cos⁡ 2α
⇒ p = |k/2 cos ⁡α| ⇒ 2p = |k cos⁡ 2α| .... (i)
second line is x sin⁡ α + y cos⁡ α = k sin 2α
⇒ q = |k sin ⁡2α| .... (ii)
Hence, 4p2 + q2 = k2 (From (i) & (ii))


Q.31. The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m − n = 0 and mn + nl + lm = 0, is :        (JEE Main 2021)
(a) π/2
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c)JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) π/3

Ans. a

n = 2 (l + m)
lm + n(l + m) = 0
lm + 2(l + m)2 = 0
2l2 + 2m2 + 5ml = 0
2(l/m)+ 2 + 5(l/m) = 0
2t2 + 5t + 2 = 0
(t + 2)(2t + 1) = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(i) l/m = −2
(−2m, m, −2m)
(−2, 1, −2)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
n = −2l
(l, −2l, −2l)
(1, −2, −2)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.32. Let A be a fixed point (0, 6) and B be a moving point (2t, 0). Let M be the mid-point of AB and the perpendicular bisector of AB meets the y-axis at C. The locus of the mid-point P of MC is :        (JEE Main 2021)
(a) 3x2 − 2y − 6 = 0
(b) 3x+ 2y − 6 = 0
(c) 2x2 + 3y − 9 = 0
(d) 2x2 − 3y + 9 = 0

Ans. c

A(0, 6) and B(2t, 0)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Perpendicular bisector of AB is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let P be (h, k)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.33. Let ABC be a triangle with A(−3, 1) and ∠ACB = θ, 0 < θ < π/2. If the equation of the median through B is 2x + y − 3 = 0 and the equation of angle bisector of C is 7x − 4y − 1 = 0, then tanθ is equal to:        (JEE Main 2021) 
(a) 1/2
(b) 3/4
(c) 4/3
(c) 2

Ans. c
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE lies on 2x + y − 3 = 0
⇒ 2a + b = 11 ...........(i)
∵ C lies on 7x − 4y = 1
⇒ 7a − 4b = 1 ......... (ii)
∴ by (i) and (ii) : a = 3, b = 5
⇒ C(3, 5)
∴ mAC = 2/3
Also, mCD = 7/4
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.34. Let the equation of the pair of lines, y = px and y = qx, can be written as (y − px) (y − qx) = 0. Then the equation of the pair of the angle bisectors of the lines x2 − 4xy − 5y2 = 0 is :        (JEE Main 2021)
(a) x2 − 3xy + y2 = 0
(b) x2 + 4xy − y2 = 0
(c) x2 + 3xy − y2 = 0
(d) x2 − 3xy − y2 = 0

Ans. c
Equation of angle bisector of homogeneous equation of pair of straight line ax2 + 2hxy + by2 is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
for x2 – 4xy – 5y2 = 0
a = 1, h = – 2, b = – 5
So, equation of angle bisector is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ x2 − y2 = −3xy
So, combined equation of angle bisector is x2 + 3xy - y2 = 0


Q.35. Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :        (JEE Main 2021)
(a) (1, 2)
(b) (2, 2)
(c) (2, 1)
(d) (1, 3)

Ans. b
Both the lines pass through origin.

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
point D is equal to intersection of 4x + 5y = 0 & 11x + 7y = 9
So, coordinates of point JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Also, point B is point of intersection of 7x + 2y = 0 and 11x + 7y = 9
So, coordinates of point JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
diagonals of parallelogram intersect at middle let middle point of B, D
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
equation of diagonal AC
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
y = x
diagonal AC passes through (2, 2)


Q.36. The point P (a, b) undergoes the following three transformations successively :
(A) reflection about the line y = x.
(B) translation through 2 units along the positive direction of x-axis.
(C) rotation through angle π/4 about the origin in the anti-clockwise direction.
If the co-ordinates of the final position of the point P are JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE, then the value of 2a + b is equal to :        (JEE Main 2021)
(a) 13
(b) 9
(c) 5
(d) 7

Ans. b
Image of A(a, b) along y = x is B(b, a). Translating it 2 units it becomes C(b + 2, a).
Now, applying rotation theorem
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ b − a + 2 = −1 ......(i)
and b + 2 + a = 7 ...... (ii)
⇒ a = 4; b = 1
⇒ 2a + b = 9


Q.37. Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x + y = 3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :        (JEE Main 2021)
(a) 7√2
(b) 9/√2
(c) 2√2
(d) 3√2

Ans. b

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
sin⁡ 30∘ = r/R = 1/2
R = 2r
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.38. The equation of one of the straight lines which passes through the point (1, 3) and makes an angles tan−1(√2) with the straight line, y + 1 = 3√2 x is        (JEE Main 2021) 
(a) 4√2x + 5y−(15 + 4√2) = 0
(b) 5√2x + 4y − (15 + 4√2) = 0
(c) 4√2x + 5y − 4√2 = 0
(d) 4√2x − 5y − (5 + 4√2) = 0

Ans. a
Let slope of line be m
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence line can be
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.39. The number of integral values of m so that the abscissa of point of intersection of lines 3x + 4y = 9 and y = mx + 1 is also an integer, is :        (JEE Main 2021) 
(a) 1
(b) 2
(c) 3
(d) 0

Ans. b
3x + 4(mx + 1) = 9
⇒ x(3 + 4m) = 5
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ (3 + 4m) = ±1, ±5
⇒ 4m = −3 ± 1, −3 ± 5
⇒ 4m = −4, −2, −8, 2
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∴ Two integral value of m.


Q.40. In a triangle PQR, the co-ordinates of the points P and Q are (−2, 4) and (4, −2) respectively. If the equation of the perpendicular bisector of PR is 2x − y + 2 = 0, then the centre of the circumcircle of the ΔPQR is :        (JEE Main 2021)
(a) (−1, 0)
(b) (1, 4)
(c) (0, 2)
(d) (−2, −2)

Ans. d
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Slope of perpendicular bisector of PQ = 1
Equation of perpendicular bisector of PQ
y − 1 = 1(x − 1)
⇒ y = x
Solving with perpendicular bisector of PR,
2x − y + 2 = 0
Circumcentre is (−2, −2)


Q.41. Let A(−1, 1), B(3, 4) and C(2, 0) be given three points.
A line y = mx, m > 0, intersects lines AC and BC at point P and Q respectively. Let A1 and A2 be the areas of ΔABC and ΔPQC respectively, such that A1 = 3A2, then the value of m is equal to :        (JEE Main 2021) 

(a) 1
(b) 3
(c) 2
(d) 415

Ans. a
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
A1 = 13/2
Equation of line AC is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Line AC intersect with line y = mx at P,
Solving we get JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Equation of line BC is y − 0 = 4(x − 2)
Line BC intersect with line y = mx at Q,
Solving we get JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒12m= ±(3m−11m − 4)
taking +ve sign
9m2 + 11m + 4 = 0 (Rejected ∵ m is imaginary)
taking −ve sign
15m2 − 11m − 4 = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∴ m = 1 (As given m > 0)


Q.42. The intersection of three lines x − y = 0, x + 2y = 3 and 2x + y = 6 is a :        (JEE Main 2021)
(a) Right angled triangle
(b) Equilateral triangle
(c) None of the above
(d) Isosceles triangle

Ans. d

The given three lines are x − y = 0, x + 2y = 3 and 2x + y = 6 then point of intersection,
lines x − y = 0 and x + 2y = 3 is (1, 1)
lines x − y = 0 and 2x + y = 6 is (2, 2)
and lines x + 2y = 3 and 2x + y = 0 is (3, 0)
The triangle ABC has vertices A(1, 1), B(2, 2) and C(3, 0)
∴ AB = √2, BC = √5 and AC = √5
∴ ΔABC is isosceles


Q.43. The image of the point (3, 5) in the line x − y + 1 = 0, lies on :        (JEE Main 2021)
(a) (x − 4)2 + (y − 4)2 = 8
(b) (x − 4)2 + (y + 2)2 = 16
(c) (x − 2)2 + (y − 2)2 = 12
(d) (x − 2)2 + (y − 4)2 = 4

Ans. d
So, let the image is (x, y)
So, we have

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ x = 4, y = 4
⇒ Point (4, 4)
Which will satisfy the curve
(x − 2)2 + (y − 4)2 = 4
as (4 − 2)2 + (4 − 4)2 = 4 + 0 = 4


Q.44. A man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts 14 of this line on the coordinate axes is Three stones A, B and C are placed at the points (1, 1), (2, 2) and (4, 4) respectively. Then which of these stones is/are on the path of the man ?        (JEE Main 2021)
(a) A only
(b) All the three
(c) C only
(d) B only

Ans. d
Given, position of A = (1, 1)
Position of B = (2, 2)
Position of C = (4, 4)

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let x-intercept be a and y-intercept be b.
Equation of line traced is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
This is the equation of path, let a point (h, k) lie on this path.
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Also, AM of reciprocal of a and b = 1/4
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
On comparing Eqs. (i) and (ii), we get (h, k) = (2, 2)
Hence, the required stone is B(2, 2).


Q.45. Let A(1 , 0), B(6, 2) and C(3/2, 6) be the vertices of a triangle ABC. If P is a point inside the triangle ABC such that the triangles APC, APB and BPC have equal areas, then the length of the line segment PQ, where Q is the point (-7/6, -1/3), is ___.    (2020)

Ans. (5.00)
We have
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
If the triangles APC, APB and BPC have equal areas, then the point P must be the centroid of triangle ABC. Hence, the coordinates of point P is
P = (17/6, 8/3)
Now, the length of line segment PQ is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.46. The locus of the mid-points of the perpendicular drawn from points on the line x = 2y to the line x = y is    (2020)
(a) 2x – 3y = 0
(b) 5x – 7y = 0
(c) 3x – 2y = 0
(d) 7x – 5y = 0

Ans. b
Let the mid-point of perpendicular is (h, k).
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Now, the slope of perpendicular lines is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Now,  JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE ...(1)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE ...(2)
From Eqs. (1) and (2), we get
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, the locus of mid-point of perpendiculars is
5x = 7y ⇒ 5x - 7y = 0.


Q.47. Let two points be A (1, −1) and B (0, 2). If a point P(x', y')be such that the area of ΔPAB = 5 sq. units and it lies on the line 3x + y - 4λ = 0, then a value of λ is    (2020)
(a) 4
(b) 3
(c) 2
(d) −3

Ans. b
The equation of the line passing through the points A (1, −1) and B (0, 2) is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE ...(1)
Now,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
The lines 3x + 2y - 2 = 0 and 3x + y - 4λ = 0 are parallel lines and distance between them is equal to the height of triangle PAB. So,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE ...(2)
The area of the triangle PAB is 5 sq. units, then
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ λ = -2, 3


Q.48. Let C be the centroid of the triangle with vertices (3, −1), (1, 3) and (2, 4). Let P be the point of intersection of the lines x + 3y - 1 = 0 and 3x - y  + 1 = 0. Then the line passing through the points C and P also passes through the point:    (2020)
(a) (−9, −6)
(b) (9, 7)
(c) (7, 6)
(d) (−9, −7)

Ans. a
The coordinates of point C is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
The intersection point P of the lines x + 3y - 1 = 0 and 3x - y  + 1 = 0 is
x = -1/5 and y = 2/5
Now, the equation of line CP is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ 8x - 11y + 6 = 0
Hence, from the given options , the line passes through the point (−9, −6).
⇒ 8(-9) + 66 + 6 = 0


Q.49. Consider the set of all lines px + qy + r = 0 such that 3p + 2q + 4r = 0. Which one of the following statements is true?    (2019)
(a) The lines are concurrent at the point JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

(b) Each line passes through the origin.
(c) The lines are all parallel.
(d) The lines are not concurrent.

Ans. a
The given equations of the set of all lines
px + qy + r = 0    ...(1)
and given condition is:
3p + 2q + 4r = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE   ...(2)
From (1) & (2) we get:
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence the set of lines are concurrent and passing through the fixed point JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.50. Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is:    (2019)
(a) 9    
(b) 18
(c) 36    
(d) 32

Ans. c
One of the possible ΔOAB is A(a, 0) and B(0, b).
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
no. of integral solutions of the equation is the number of
 divisors of 100 : 1, 2, 4, 5, 10, 20, 25, 50, 100 = Total divisors = 9  
So, (9 × 4) = 36


Q.51. Let the equations of two sides of a triangle be 3x - 2y + 6 = 0 and 4x + 5y - 20 = 0. If the orthocentre of this triangle is at (1, 1), then the equation of its third side is:    (2019)
(a)  122y - 26x - 1675 = 0
(b) 122y + 26x + 1675 = 0
(c) 26x + 61y + 1675 = 0
(d) 26x - 122y - 1675 = 0

Ans. d
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Since, AH is perpendicular to BC
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.52. A point P moves on the line 2x - 3y + 4 = 0. If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of ΔPQR is a line:    (2019)
(a) with slope 3/2
(b) parallel to x-axis
(c) with slope 2/3
(d) parallel to y-axis

Ans. c
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE 

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.53. If the line 3x + 4y - 24 = 0 intersects the x-axis at the point A and the x-axis at the point B, then the incentre of the triangle OAB, where O is the origin, is:    (2019)
(a) (3,4)    
(b) (2,2)
(c) (4,3)    
(d) (4,4)

Ans. b
Equation of the line is:
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.54. Two sides of a parallelogram are along the lines, x + y = 3 and x - y + 3 = 0. If its diagonals intersect at (2, 4), then one of its vertex is:    (2019)
(a) (3,5)    
(b) (2,1)
(c) (2,6)    
(d) (3,6)

Ans. d
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Since, x - y + 3 = 0 and x + y = 3 are perpendicular lines and intersection point of x - y + 3 = 0 and x + y = 3 is P(0,3).
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Then, the vertex D is (3, 6).


Q.55. Two vertices of a triangle are (0, 2) and (4, 3). If its orthocentre is at the origin, then its third vertex lies in which quadrant?    (2019)
(a) third    
(b) second
(c) first    
(d) fourth

Ans. b
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, vertex R lies in second quadrant.


Q.56. The straight line x + 2y = 1 meets the coordinate axes at A and B. A circle is drawn through A, B and the origin. Then the sum of perpendicular distances from A and B on the tangent to the circle at the origin is:    (2019)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE    
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE   
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. a 
Equation of circle
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.57. If in a parallelogram ABDC, the coordinates of A, B and C are respectively (1, 2), (3,4) and (2,5), then the equation of the diagonal AD is:    (2019)
(a) 5x - 3y + 1=0    
(b) 5x + 3y - 11=0
(c) 3x - 5y + 7 = 0    
(d) 3x + 5y - 13 = 0

Ans. a
Since, in parallelogram mid points of both diagonals coincides.
∴ mid-point of AD = mid-point of BC
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
=JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.58. If the straight line, 2x - 3y + 17 = 0 is perpendicular to the line passing through the points (7, 17) and (15, β), then β equals:    (2019)
(a) 35/3    
(b) -5
(c) -35/3 
(d) 5

Ans. d
∵ Equation of straight line can be rewritten as,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∴ Slope of straight line = 2/3
Slope of line passing through the points (7, 17) and
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Since, lines are perpendicular to each other.
Hence, m1m2 = -1
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.59. If a straight line passing through the point P(-3, 4) is such that its intercepted portion between the coordinate axes is bisected at P, then its equation is:    (2019)
(a) 3x - 4y + 25 = 0    
(b) 4x - 3y + 24 = 0
(c) x - y + 7 = 0    
(d) 4x + 3y = 0

Ans. b
Since, P is mid point of MN
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ y = 8
Hence required equation of straight line MN is
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ 4x - 3y + 24 = 0


Q.60. A point on the straight line, 3x + 5y =15 which is equidistant from the coordinate axes will lie only in:    (2019)
(a) 4th quadrant
(b) 1st quadrant
(c) 1st and 2nd quadrants
(d) 1st, 2nd and 4th quadrants

Ans. c
A point which is equidistant from both the axes lies on either y = x and y = -x.
Since, point lies on the line 3x + 5y = 15
Then the required point
3x + 5y = 15
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, the required point lies in 1st and 2nd quadrant.

Q.61. Let 0(0, 0) and A(0, 1) be two fixed points. Then the locus of a point P such that the perimeter of ΔAOP is 4, is:    (2019)
(a) 8x2 - 9y2 + 9y = 18
(b) 9x2 - 8y2+ 8y = 16
(c) 9x2 + 8y2 - 8y = 16
(d) 8x2 + 9y2 - 9y = 18

Ans. c
Let point P (h, k)
∵ OA = 1
So, OP + AP = 3
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.62. Suppose that the points (h, k), (1, 2) and (-3, 4) lie on the line L1. If a line L2 passing through the points (h, k) and (4, 3) is perpendicular on L1, then k/h equals:    (2019)
(a) 1/3    
(b) 0    
(c) 3    
(d) -1/7

Ans. a
∵ (h, k), (1,2) and (- 3, 4) are collinear
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ h + 2k = 5   ...(i)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
By the given points (h, k) and (4, 3),
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.63. Slope of a line passing through P(2, 3) and intersecting the line x + y = 7 at a distance of 4 units from P, is:    (2019)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. b
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Since point at 4 units from P (2, 3) will be
A (4 cosθ + 2, 4 sin (θ + 3) and this point will satisfy the equation of line x + y = 7
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.64. If the two lines x + (a - 1)y = 1 and 2x + a2y =1 (a ∈ R - {0,1}) are perpendicular, then the distance of their point of intersection from the origin is:    (2019)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) 2/5
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. a
∵ two lines are perpendicular ⇒ m1m2 = - 1
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence equations of lines are x - 2y= 1 and 2x + y = 1
∴ intersection point is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Now, distance from originJEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.65. A rectangle is inscribed in a circle with a diameter lying along the line 3y = x + 7. If the two adjacent vertices of the rectangle are (-8, 5) and (6, 5), then the area of the rectangle (in sq. units) is:    (2019)
(a) 84    
(b) 98    
(c) 72    
(d) 56

Ans. a
Given situation
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
∵ perpendicular bisector of AB will pass from centre.
∴ equation of perpendicular bisector x = -1
Hence centre of the circle is (- 1, 2)
Let co-ordinate of D is (α, β)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ α = - 8 and β = - 1, D ≡ (-8, -1)
|AD| = 6 and |AB| = 14
Area = 6 x 14 = 84 sq.units


Q.66. Lines are drawn parallel to the line 4x - 3y + 2 = 0, at a distance 3/5 from the origin. Then which one of the following points lies on any of these lines ?    (2019)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. a
Let straight line be 4x - 3y + α = 0
∵ distance from origin = 3/5
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, line is 4x - 3y + 3 = 0 or 4x - 3y - 3 = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.67. The equation y = sin x sin (x + 2) - sin2 (x + 1) represents a straight line lying in:    (2019)
(a) second and third quadrants only
(b) first, second and fourth quadrant
(c) first, third and fourth quadrants
(d) third and fourth quadrants only

Ans. d
Consider the equation,
y = sin x. sin (x + 2) - sin2(x + 1)
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
By the graph y lies in III and IV quadrant.


Q.68. A triangle has a vertex at (1, 2) and the mid points of the two sides through it are (-1, 1) and (2, 3). Then the centroid of this triangle is:    (2019)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. b
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.69. A straight line L at a distance of 4 units from the origin makes positive intercepts on the coordinate axes and the perpendicular from the origin to this line makes an angle of 60° with the line x + y = 0. Then an equation of the line L is:    (2019)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. b, d
∵ Perpendicular makes an angle of 60° with the line x + y = 0.
∴ The perpendicular makes an angle of 15° or 75° with x-axis.
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, the equation of line will be
x cos 75° + y sin 75° = 4
or x cos 15° +y sin 15° = 4
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.70. A straight line through a fixed point (2, 3) intersects the coordinate axes at distinct points P and Q. If O is the origin and the rectangle OPRQ is completed, then the locus of R is:    (2018)
(a) 3x + 2y = 6
(b) 2y + 3x = xy
(c) 3x + 2y = xy
(d) 3x + 2y = 6xy

Ans. b
Let R = (h,k)
P = (0, k)
Q = (h,0)
Equation of line would be,
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
2k + 3h = hk
Locus of (h, k) is 2y + 3x = xy


Q.71. Let k be an integer such that the triangle with vertices (k, –3k), (5, k) and (–k, 2) has area 28 sq. units. Then the orthocentre of this triangle is at the point    (2017)
(a) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(b) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. a
We have
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
or
5k2 + 13k -46 = 0 (no real solution exist)
∴ k = -23/5
or k = 2
k is an integer, so k=2
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Therefore (2, 1/2)


Q.72. A square, of each side 2, lies above the x-axis and has one vertex at the origin. If one of the sides passing through the origin makes an angle 30° with the positive direction of the x-axis, then the sum of the x-coordinates of the vertices of the square is:    (2017)
(a) 2√3-2
(b) √3-2
(c) 2√3-1
(d) √3-1
Ans.
a
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.73. Two sides of a rhombus are along the lines, x - y + 1 = 0 and 7x - y - 5 = 0. If its diagonals intersect at (-1, -2), then which one of the following is a vertex of this rhombus?    (2016)
(a) (-3, -9)
(b) (-3, -8)
(c) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
(d) JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE

Ans. c
Coordinates of A = (1, 2)
∴ Slope of AE = 2
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ Slope of BD = JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ Eq. of BD is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ x + 2y + 5 = 0
Co-ordinates of D = JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.74. If a variable line drawn through the intersection of the lines JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE and JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE meets the coordinate axes at A and B, (A ≠ B), then the locus of the midpoint of AB is    (2016)
(a) 7xy = 6(x + y)
(b) 6xy = 7(x + y)
(c) 4(x + y)2 – 28(x + y) + 49 = 0
(d) 14(x + y)2 – 97 (x + y) + 168 = 0 

Ans. a
4x + 3y = 12 ....(1)
3x + 4y = 12 ....(2)
equation of lines passing through the intersection of the lines
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
4x + 3y – 12 + λ(3x + 4y - 12) = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Hence, locus is 7xy = 6(x + y)


Q.75. The point (2, 1) is translated parallel to the line L : x – y = 4 by JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE units. If the new point Q lies in the third quadrant, then the equation of the line passing through Q and perpendicular to L is:    (2016)
(a) 2x + 2y = 1 - √6
(b) x = y = 3 - 3 √6
(c) x + y = 2 - √6
(d) x + y =  3 - 2 √6

Ans. d
Slopes of x – y = 4
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒ Equation of required line is JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.76. A straight line through origin O meets the line 3y = 10 – 4x and 8x + 6y + 5 = 0 at points A and B respectively. Then O divides the segment AB in the ratio    (2016)
(d) 3 : 4
(b) 1 : 2
(c) 2 : 3
(d) 4 : 1

Ans. d
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE


Q.77. A ray of light is incident along a line which meets another line, 7x – y + 1 = 0, at the point (0, 1). The ray is then reflected from this point along the line, y + 2x = 1. Then the equation of the line of incidence of the ray of light is    (2016)
(a) 41x + 38y – 38 = 0
(b) 41x – 38y + 38 = 0
(c) 41x + 25y – 25 = 0
(d) 41x – 25y + 25 = 0 

Ans. b
Incident line
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
Let a point (1, –1) on  y + 2x = 1
And image of (1, –1) lie on incident line in
7x – y + 1 = 0
JEE Main Previous year questions (2016-22): Straight Lines and Pair of Straight Lines Notes | Study Mathematics For JEE - JEE
⇒369x –342y + 342 = 0
⇒41x –38y + 38 = 0

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