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# JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q.1. Let ABC be the triangle with AB = 1, AC = 3 and ∠BAC = π/2. If a circle of radius r > 0 touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of r is __________.         (JEE Advanced 2022)

Ans. Between 0.82 and 0.86

Here ABC is a right angle triangle. BC is the Hypotenuse of the triangle.

We know, diameter of circumcircle of a right angle triangle is equal to the Hypotenuse of the triangle also midpoint of Hypotenuse is the center of circle.

∴ BC = Diameter of the circle

Here B = (0, 1) and C = (3,0)

Radius of circumcircle (R) = √10/2

∴ Center of circle (M) =

Center of circle which touches line AB and AC = (r, r)

Now distance between center of two circles,

⇒ r2 − 4r + √10r = 0

⇒ r(r − 4 + √10) = 0

⇒ r = 0 or r = r − √10

∴ r = 4 − √10 [as r ≠ 0]

= 0.837
≃ 0.842

Q.2. Let G be a circle of radius R > 0. Let G1, G2,…, Gn be n circles of equal radius r > 0. Suppose each of the n circles G1, G2,…, Gn touches the circle G externally. Also, for i = 1, 2,…, n − 1, the circle Gtouches Gi+1 externally, and Gn touches G1 externally. Then, which of the following statements is/are TRUE?         (JEE Advanced 2022)
(a) If n = 4, then (√2 − 1)r < R
(b) If n = 5, then r < R
(c) If n = 8, then (√2 − 1)r < R
(d) If n = 12, then √2(√3 + 1)r > R

Ans. c, d

Here if we add center of circles G1, G2, G3 ....... Gn, then we get a polygon of n sides.

From figure you can see one side of polygon makes angle θ with the center.

∴ n sides make angle = nθ

We know, nθ = 2π

⇒ θ = 2π/n

Here triangle OMN is an isosceles triangle. Line joining of point O and midpoint O of MN (point A) is perpendicular to line MN and perpendicular bisector of angle θ.

∴ ∠MOA = θ/2 = π/n

From right angle triangle OMA,

Option A:

When n = 4,

∴ Option (A) is wrong.

Option B:

When n = 5,

We know, in 0 to π/2, cosecθ is decreasing.

∴ cosec45∘ < cosec36 < cosec30

∴ R < r, when n = 5

∴ Option (B) is wrong.

Option C:

When n = 8,

cosecθ is decreasing function in 0 to π/2.

∴ Option (C) is correct.

Option D:

When n = 12, then

Option (D) is correct.

Q.3. Let Let S = {(x, y) ∈ N × N : 9(x − 3)2 + 16(y − 4)2 ≤ 144} and T = {(x, y) ∈ R × R : (x − 7)2 + (y − 4)2 ≤ 36}. Then n(S ∩ T) is equal to __________.         (JEE Main 2022)

Ans. 27

Q.4. Let AB be a chord of length 12 of the circle (x − 2)2 + (y + 1)2 = 169/4. If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to __________.         (JEE Main 2022)

Ans. 72
Here AM = BM = 6

sin⁡θ = 12/13
In △PAO:
PO/OA = sec⁡θ

Q.5. Let the mirror image of a circle c: x2 + y2 − 2x − 6y + α = 0 in line y = x + 1 be c2 : 5x2 + 5y2 + 10gx + 10fy + 38 = 0. If r is the radius of circle c2, then α + 6r2 is equal to ________.         (JEE Main 2022)

Ans. 12
c: x2 + y2 − 2x − 6y + α = 0

Then centre = (1, 3) and radius (r) =

Image of (1, 3) w.r.t. line x − y + 1 = 0 is (2, 2)

c2 : 5x2 + 5y2 + 10gx + 10fy + 38 = 0

Then (−g, −f) = (2, 2)

∴ g = f = −2 .......... (i)

Q.6. If the circles x2 + y2 + 6x + 8y + 16 = 0 and x+ y2 + 2(3 − √3)x + 2(4 − √6)y = k + 6√3 + 8√6, k > 0, touch internally at the point P(α, β), then (α + √3)2 + (β + √6)2 is equal to ________________.         (JEE Main 2022)

Ans. 25

The circle x2 + y2 + 6x + 8y + 16 = 0 has centre (−3,−4) and radius 3 units.

The circle x+ y2 + 2(3 − √3)x + 2(4 − √6)y = k + 6√3 + 8√6, k > 0 has centre
These two circles touch internally hence

Here, k = 2 is only possible (∵ k > 0)

Equation of common tangent to two circles is 2√3x + 2√6y + 16 + 6√3 + 8√6 + k = 0

∵ k = 2 then equation is

x + √2y + 3 + 4√2 + 3√3 = 0 ...... (i)

∵ (α, β) are foot of perpendicular from (−3, −4)

To line (i) then

⇒ (α + √3)2 = 9 and (β + √6)2 = 16
∴ (α + √3)2 + (β + √6)2 = 25

Q.7. If one of the diameters of the circle x2 + y2 − 2√2x − 6√2y + 14 = 0 is a chord of the circle (x−2√2)2 + (y − 2√2)2 = r2, then the value of r2 is equal to ____________.         (JEE Main 2022)

Ans. 10

Q.8. Let the lines y + 2x = √11 + 7√7 and 2y + x = 2√11 + 6√7 be normal to a circle C : (x − h)2 + (y − k)2 = r2. If the line √11y − 3x =is tangent to the circle C, then the value of (5h − 8k)+ 5r2 is equal to __________.         (JEE Main 2022)

Ans. 816

L1: y + 2x = √11 + 7√7

L2: 2y + x = 2√11 + 6√7

Point of intersection of these two lines is centre of circle i.e.

r from centre to lineis radius of circle

So (5h − 8K)2 + 5r2

= 64 × 11 + 112 = 816

Q.9. Let a circle C of radius 5 lie below the x-axis. The line L1 : 4x + 3y + 2 = 0 passes through the centre P of the circle C and intersects the line L2 = 3x − 4y − 11 = 0 at Q. The line L2 touches C at the point Q. Then the distance of P from the line 5x − 12y + 51 = 0 is ______________.         (JEE Main 2022)

Ans. 11
L1: 4x + 3y + 2 = 0

L2: 3x − 4y − 11 = 0

Since circle C touches the line L2 at Q intersection point Q of L1 and L2, is (1, −2)

∵ P lies of L1

⇒ (x − 1)2 = 9

⇒ x = 4, −2

∵ Circle lies below the x-axis

∴ y = −6

P(4, −6)

Now distance of P from 5x − 12y + 51 = 0

Q.10. A rectangle R with end points of one of its sides as (1, 2) and (3, 6) is inscribed in a circle. If the equation of a diameter of the circle is 2x − y + 4 = 0, then the area of R is ____________.         (JEE Main 2022)

Ans. 16

As slope of line joining (1, 2) and (3, 6) is 2 given diameter is parallel to side

Q.11. Let the abscissae of the two points P and Q be the roots of 2x2 − rx + p = 0 and the ordinates of P and Q be the roots of x2 − sx − q = 0. If the equation of the circle described on PQ as diameter is 2(x2 + y2) − 11x − 14y − 22 = 0, then 2r + s − 2q+p is equal to __________.         (JEE Main 2022)

Ans. 7
Let P(x1, y1) & Q(x2, y2)

∴ Roots of 2x2 − rx + p = 0 are x1, x2

and roots of x− sx − q = 0 are y1, y2.

∴ Equation of circle ≡ (x − x1)(x − x2) + (y − y1)(y − y2) = 0

⇒ x2 − (x1 + x2)x + x1x2 + y2 − (y1 + y2)y + y1y= 0

⇒ 2x2 + 2y2 − rx + 2sy + p − 2q = 0

Compare with 2x2 + 2y2 − 11x − 14y − 22 = 0

We get r = 11, s = 7, p − 2q = −22
⇒ 2r + s + p − 2q = 22 + 7 − 22 = 72

Q.12. Let a circle C : (x − h)2 + (y − k)2 = r2, k > 0, touch the x-axis at (1, 0). If the line x + y = 0 intersects the circle C at P and Q such that the length of the chord PQ is 2, then the value of h + k + r is equal to ___________.         (JEE Main 2022)

Ans. 7

Here, OM2 = OP2 − PM2

∴ r2 − 2r − 3 = 0

∴ r = 3

∴ Equation of circle is

(x − 1)2 + (y − 3)2 = 32

∴ h = 1, k = 3, r = 3

∴ h + k + r = 7

Q.13. Let the tangents at two points A and B on the circle x2 + y2 − 4x + 3 = 0 meet at origin O(0,0). Then the area of the triangle OAB is:         (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. b
x2 + y2 − 4x + 3 = 0

⇒ (x − 2)2 + y2 = 1

Also, AO = BO

Q.14. The foot of the perpendicular from a point on the circle x2 + y2 = 1, z = 0 to the plane 2x + 3y + z = 6 lies on which one of the following curves?         (JEE Main 2022)
(a) (6x + 5y − 12)+ 4(3x + 7y − 8)2 = 1, z = 6 − 2x − 3y
(b) (5x + 6y − 12)2 + 4(3x + 5y − 9)2 = 1, z = 6 − 2x − 3y
(c) (6x + 5y − 14)2 + 9(3x + 5y − 7)2 = 1, z = 6 − 2x − 3y
(d) (5x + 6y − 14)+ 9(3x + 7y − 8)= 1, z = 6 − 2x − 3y

Ans. b
Any point on x2 + y2 = 1, z = 0 is p(cos⁡θ, sin⁡θ, 0)

If foot of perpendicular of p on the plane 2x + 3y + z = 6 is (h, k, l) then

h = 2r + cos⁡θ, k = 3r + sin⁡θ, l = r

Hence, h − 2l = cos⁡θ and k − 3l = sin⁡θ

Hence, (h − 2l)2 + (k − 3l)2 = 1

When l = 6 − 2h − 3k

Hence required locus is

(x − 2(6 − 2x − 3y))+ (y − 3(6 − 2x −3y))2 = 1

⇒ (5x + 6y − 12)2 + 4(3x + 5y − 9)= 1, z = 6 − 2x − 3y

Q.15. Let C be the centre of the circle x2 + y2 − x + 2y = 11/4 and P be a point on the circle. A line passes through the point C, makes an angle of π/4 with the line CP and intersects the circle at the points Q and R. Then the area of the triangle PQR (in unit 2 ) is:         (JEE Main 2022)
(a) 2
(b) 2√2
(c) 8sin⁡(π/8)
(d) 8cos⁡(π/8)

Ans. b

QR = 2r = 4

= 2√2 sq. units

Q.16. For t ∈ (0, 2π), if ABC is an equilateral triangle with vertices A(sin⁡t, −cos⁡t), B(cost, sin⁡t) and C(a, b) such that its orthocentre lies on a circle with centre (1, 1/3), then (a2 − b2) is equal to:        (JEE Main 2022)
(a) 2
(a) 8/3
(b) 8
(c) 77/9
(d) 80/9

Ans. b
Let P(h, k) be the orthocentre of ΔABC

Then

(orthocentre coincide with centroid)

∴ (3h − a)2 + (3k − b)2 = 2

∵ orthocentre lies on circle with centre (1, 1/3)

∴ a = 3, b = 1

∴ a− b2 = 8

Q.17. A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of circle C1. Let Cbe the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to:       (JEE Main 2022)
(a) 1 : 4
(b) 1 : 5
(c) 2 : 5
(d) 1 : 3

Ans. a

Equation of C1

x+ y− 4x = 0

Intersection with

y = 2x

x+ 4x− 4x = 0

5x2 − 4x = 0

x + 2y = 4 ⇒ P : (4, 0), Q : (0, 2)
QA : AP = 1 : 4

Q.18. If the circle x2 + y2 − 2gx + 6y − 19c = 0, g, c ∈ R passes through the point (6, 1) and its centre lies on the line x − 2cy = 8, then the length of intercept made by the circle on x-axis is       (JEE Main 2022)
(a) √11
(b) 4
(c) 3
(d) 2√23

Ans. d

Circle: x2 + y2 − 2gx + 6y − 19c = 0

It passes through h(6, 1)

⇒ 36 + 1 − 12g + 6 − 19c = 0

= 12g + 19c = 43 ..... (1)

Line x − 2cy = 8 passes through centre

⇒ g + 6c = 8 ...... (2)

From (1) & (2)

g = 2, c = 1

C: x2 + y2 − 4x + 6y − 19 = 0

Q.19. Let the abscissae of the two points P and Q on a circle be the roots of x− 4x − 6 = 0 and the ordinates of P and Q be the roots of y2 + 2y − 7 = 0. If PQ is a diameter of the circle x2 + y2 + 2ax + 2by + c = 0, then the value of (a + b − c) is _____________.       (JEE Main 2022)
(a) 12
(b) 13
(c) 14
(d) 16

Ans. a
Abscissae of PQ are roots of x− 4x − 6 = 0

Ordinates of PQ are roots of y2 + 2y − 7 = 0

and PQ is diameter

⇒ Equation of circle is

x2 + y2 − 4x + 2y − 13 = 0

But, given x2 + y2 + 2ax + 2by + c = 0

By comparison a = −2, b = 1, c = −13
⇒ a + b − c = −2 + 1 + 13 = 12

Q.20. Consider three circles:
C1: x2 + y= r2
C2: (x − 1)2 + (y − 1)2 = r2
C3: (x − 2)+ (y − 1)2 = r2
If a line L : y = mx + c be a common tangent to C1, C2 and C3 such that C1 and C3 lie on one side of line L while C2 lies on other side, then the value of 20(r2 + c) is equal to:       (JEE Main 2022)
(a) 23
(b) 15
(c) 12
(d) 6

Ans. d

C1: x2 + y= r2 ; center = (0, 0) and radius = r

C2: (x − 1)2 + (y − 1)2 = r2 ; center = (1, 1) and radius = r

C3: (x − 2)+ (y − 1)2 = r2 ; center = (2, 1) and radius = r

Distance of y = mx + c line from center (0, 0) is,

Distance of y = mx + c line from center (1, 1) is,

Distance of y = mx + c line from center (2, 1) is,

From (1) and (2), we get

⇒ m − 1 + c = ±c ..... (4)

taking positive sign,

m − 1 + c = c

⇒ m − 1 = 0

⇒ m = 1

From (2) and (3), we get

⇒ (m − 1 + c) = ±(2m − 1 + c) ...... (5)

taking positive sign,

m − 1 + c = 2m − 1 + c

⇒ m = 0

By taking positive sign we get two different value of m so it is not acceptable.

From equation (4), taking negative sign,

m − 1 + c = −c

⇒ m + 2c − 1 = 0 ..... (6)

From equation (5), taking negative sign

m − 1 + c = −(2m − 1 + c)

⇒ 3m + 2c − 2 = 0 ..... (7)

Solving equation (6) and (7), we get

3m + 1 − m − 2 = 0

⇒ 2m = 1

⇒ m = 1/2

⇒ c = 1/4

Putting value of m = 1/2 and c = 1/4 in equation (1), we get

∴ 20(r2 + c)

= 6

Q.21. Let a triangle ABC be inscribed in the circle x2 − √2(x + y) + y2 = 0 such that ∠BAC = π/2. If the length of side AB is √2, then the area of the ΔABC is equal to:       (JEE Main 2022)
(a) 1
(b) (√6 + √3)/2
(c) (3 + √3)/4
(d) (√6 + 2√3)/4

Ans. a
Note:

For equation of circle x2 + y2 + 2gx + 2fy + c = 0, center is (−g, −f) and radius

Given,

equation of circle is

As AB and AC makes an angle 90∘ then line BC passes through the center of circle and BC is the diameter of the circle.

∴ Length of BC = 2r = 2 × 1 = 2

∴ AC2 = BC2 − AB2

= 22 − (√2)2

= 2

⇒ AC = 2

∴ Area of right angle triangle ABC

= 1 square unit.

Q.22. Let the tangent to the circle C1 : x2 + y= 2 at the point M(−1, 1) intersect the circle C2 : (x − 3)2 + (y − 2)= 5, at two distinct points A and B. If the tangents to C2 at the points A and B intersect at N, then the area of the triangle ANB is equal to:       (JEE Main 2022)
(a) 1/2
(b) 2/3
(c) 1/6
(d) 5/3

Ans. c

Equation of tangent at point M is

T = 0

⇒ xx1 + yy1 = 2

⇒ −x + y = 2

⇒ y = x + 2

Putting this value to equation of circle C2,

(x − 3)2 + (y − 2)2 = 5

⇒ (x − 3)2 + x2 = 5

⇒ x2 − 6x + 9 + x2 = 5

⇒ 2x2 − 6x + 4 = 0

⇒ x− 3x + 2 = 0

⇒ (x − 2)(x − 1) = 0

⇒ x = 1, 2

when x = 1, y = 3

and when x = 2, y = 4

∴ Point A(1, 3) and B(2, 4)

Now, equation of tangent at A(1, 3) on circle (x − 3)+ (y − 2)= 5 or x2 + y2 − 6x − 4y + 8 = 0 is

T = 0

xx1 + yy1 + g(x + x1) + f(y + y1) + C = 0

⇒ x + 3y − 3(x + 1) − 2(y + 3) + 8 = 0

⇒ x + 3y − 3x − 3 − 2y − 6 + 8 = 0

⇒ −2x + y − 1 = 0

⇒ 2x − y + 1 = 0 ....... (1)

Similarly tangent at B(2, 4) is

2x + 4y − 3(x + 2) − 2(y + 4) + 8 = 0

⇒ 2x + 4y − 3x − 6 − 2y − 8 + 8 = 0

⇒ −x + 2y − 6 = 0

⇒ x − 2y + 6 = 0 ...... (2)

Solving equation (1) and (2), we get

x − 2(2x + 1) + 6 = 0

⇒ x − 4x − 2 + 6 = 0

⇒ −3x + 4 = 0

⇒ x = 4/3

Now area of the triangle ANB

Q.23. If the tangents drawn at the points O(0, 0) and P(1 + √5, 2) on the circle x+ y− 2x − 4y = 0 intersect at the point Q, then the area of the triangle OPQ is equal to       (JEE Main 2022)
(a)
(b)
(c)
(d)

Ans. c

Q.24. The set of values of k, for which the circle C: 4x2 + 4y− 12x + 8y + k = 0 lies inside the fourth quadrant and the point (1,) lies on or inside the circle C, is       (JEE Main 2022)
(a) an empty set
(b) (6, 65/9]
(c) [80/9, 10)
(d) (9, 92/9]

Ans. d
C: 4x2 + 4y− 12x + 8y + k = 0
(1, ) lies on or inside the C

Now, circle lies in 4th quadrant centre ≡ (3/2, −1)

Q.25. Let C be a circle passing through the points A(2, −1) and B(3, 4). The line segment AB s not a diameter of C. If r is the radius of C and its centre lies on the circle (x − 5)2 + (y − 1)2 = 13/2, then r2 is equal to:       (JEE Main 2022)
(a) 32
(b) 65/2
(c) 61/2
(d) 30

Ans. b
Equation of perpendicular bisector of AB is

Solving it with equation of given circle,

But x ≠ 5/2 because AB is not the diameter.

So, centre will be (15/2, 1/2)

= 65/2

Q.26. A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is:       (JEE Main 2022)
(a) y = √2x
(b) x = √2y
(c) y2 − x2 = 2xy
(d) x2 − y2 = 2xy

Ans. d
Let the centre be (h, k)

⇒ 2h2 = h+ k2 + 2hk

Locus will be x2 − y2 = 2xy

Q.27. Let a, b and c be the length of sides of a triangle ABC such thatIf r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of R/r is equal to:       (JEE Main 2022)
(a) 5/2
(b) 2
(c) 3/2
(d) 1

Ans. a

a + b = 7λ

b + c = 8λ

c + a = 9λ

a + b + c = 12λ

∴ a = 4λ, b = 3λ, c = 5λ

Q.28. Consider the region R = {(x, y)  R × R : x  0 and y2   x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (αβ) be a point where the circle C meets the curve y2 = 4  x.
The value of α is ___________.       (JEE Advanced 2021)

Ans. 2.00
Given, x  0, y2  4  x
Let equation of circle be
(x  h)2 + y2 = h2 .... (i)

Solving Eq. (i) with y2 = 4  x, we get
x2  2hx + 4  x = 0
x2  x(2h + 1) + 4 = 0 .... (ii)
For touching/tangency, Discriminant (D) = 0
i.e. (2h + 1)2 = 16  2h + 1 = ± 4
2h = ± 4  1
h = 3/2, h = −5/2 (Rejected) because part of circle lies outside R. So, h = 3/2 = radius of circle (c).
Putting h = 3/2 in Eq. (ii),
x2  4x + 4 = 0  (x  2)2 = 0  x = 2
So, α = 2

Q.29. Consider the region R = {(x, y)  R × R : x  0 and y2   x}. Let F be the family of all circles that are contained in R and have centers on the x-axis. Let C be the circle that has largest radius among the circles in F. Let (αβ) be a point where the circle C meets the curve y2 = 4  x.

Ans. 1.50
Given, x  0, y2  4  x
Let equation of circle be
(x  h)2 + y2 = h2 .... (i)

Solving Eq. (i) with y2 = 4  x, we get
x2  2hx + 4  x = 0
x2  x(2h + 1) + 4 = 0 .... (ii)
For touching/tangency, Discriminant (D) = 0
i.e. (2h + 1)2 = 16  2h + 1 = ± 4
2h = ± 4  1
h = 3/2, h = −5/2 (Rejected) because part of circle lies outside R. So, h = 3/2 = radius of circle (c).

Q.30. Let M = {(x, y) ∈ R × R : x2 + y2 ≤ r2}, where r > 0. Consider the geometric progression , n = 1, 2, 3, ...... . Let S0 = 0 and for n ≥ 1, let Sn denote the sum of the first n terms of this progression. For n ≥ 1, let Cn denote the circle with center (Sn−1, 0) and radius an, and Dn denote the circle with center (Sn−1, Sn−1) and radius an.
Consider M withThe number of all those circles Dn that are inside M is
(a) 198
(b) 199
(c) 200
(d) 201

Ans. b

n  199
So, number of circles = 199

Q.31. Let M = {(x, y) ∈ R × R : x2 + y2 ≤ r2}, where r > 0. Consider the geometric progression , n = 1, 2, 3, ...... . Let S0 = 0 and for n ≥ 1, let Sn denote the sum of the first n terms of this progression. For n ≥ 1, let Cn denote the circle with center (Sn−1, 0) and radius an, and Dn denote the circle with center (Sn−1, Sn−1) and radius an.
Consider M with r=1025/513. Let k be the number of all those circles Cn that are inside M. Let l be the maximum possible number of circles among these k circles such that no two circles intersect. Then       (JEE Advanced 2021)
(a) k + 2l = 22
(b) 2k + l = 26
(c) 2k + 3l = 34
(d) 3k + 2l = 40

Ans. d

For circle Cn to be inside M.

Number of circles inside be 10 = k. Clearly, alternate circle do not intersect each other i.e. C1, C3, C5, C7, C9 do not intersect each other as well as C2, C4, C6, C8 and C10 do not intersect each other.
Hence, maximum 5 set of circles do not intersect each other.
l = 5
So, 3k + 2l = 40

Q.32. Consider a triangle Δ whose two sides lie on the x-axis and the line x + y + 1 = 0. If the orthocenter of Δ is (1, 1), then the equation of the circle passing through the vertices of the triangle Δ is       (JEE Advanced 2021)
(a) x+ y2 − 3x + y = 0
(b) x2 + y2 + x + 3y = 0
(c) x+ y2 + 2y − 1 = 0
(d) x2 + y2 + x + y = 0

Ans. b
Equation of circle passing through C(0, 0) is
x2 + y2 + 2gx + 2fy = 0 ..... (i)
Since Eq. (i), also passes through (1, 0) and (1, 2).

Then, 1  2g = 0  g = 1 / 2
and 5 + 1  4f = 0  f = 3 / 2
Equation of circumcircle is
i.e. x2 + y2 + x + 3y = 0

Q.33. Let Z be the set of all integers,
A = {(x, y) ∈ Z × Z: (x − 2)2 + y2 ≤ 4}
B = {(x, y) ∈ Z × Z : x2 + y2 ≤ 4}
C = {(x, y) ∈ Z × Z : (x − 2)+ (y − 2)2 ≤ 4}
If the total number of relation from A ∩ B to A ∩ C is 2p, then the value of p is:
(JEE Main 2021)
(a) 16
(b) 25
(c) 49
(d) 9

Ans. b

(x  2)2 + y2  4
x2 + y2  4
No. of points common in C1 & C2 is 5.
(0, 0), (1, 0), (2, 0), (1, 1), (1, 1)
Similarly in C2 & C3 is 5.
No. of relations = 25×5 = 225.

Q.34. A circle C touches the line x = 2y at the point (2, 1) and intersects the circle
C1 : x2 + y2 + 2y  5 = 0 at two points P and Q such that PQ is a diameter of C1. Then the diameter of C is:        (JEE Main 2021)
(a) 7√5
(b) 15
(c) 285
(d) 415

Ans. a
(x − 2)2 + (y − 1)2 + λ(x − 2y) = 0
C : x2 + y2 + x(λ − 4) + y(−2 −2λ) + 5 = 0
C1 : x2 + y2 + 2y − 5 = 0
S1 − S2 = 0 (Equation of PQ)
(λ − 4)x − (2λ + 4)y + 10 = 0 Passes through (0, −1)
⇒ λ = −7
C : x2 + y2 − 11x + 12y + 5 = 0

Diameter = 7√5

Q.35. If a line along a chord of the circle 4x2 + 4y2 + 120x + 675 = 0, passes through the point (30, 0) and is tangent to the parabola y2 = 30x, then the length of this chord is:         (JEE Main 2021)
(a) 5
(b) 7
(c) 5√3
(d) 35

Ans. d
Equation of tangent to y2 = 30 x

Q.36. Consider a circle C which touches the y-axis at (0, 6) and cuts off an intercept 6√5 on the x-axis. Then the radius of the circle C is equal to:         (JEE Main 2021)
(a) √53
(b) 9
(c) 8
(d) √82

Ans. b

Q.37. Let the circle S : 36x2 + 36y2  108x + 120y + C = 0 be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, x  2y = 4 and 2x  y = 5 lies inside the circle S, then:          (JEE Main 2021)
(a)
(b) 100 < C < 165
(c) 81 < C < 156
(d) 100 < C < 156

Ans. d
S : 36x2 + 36y2  108x + 120y + C = 0

Now,

C > 100 ...... (1)
Now, point of intersection of x  2y = 4 and 2x  y = 5 is (2, 1), which lies inside the circle S.
S(2, 1) < 0

C < 156 ..... (2)
From (1) & (2)
100 < C < 156 Ans.

Q.38. Let r1 and r2 be the radii of the largest and smallest circles, respectively, which pass through the point (4, 1) and having their centres on the circumference of the circle x2 + y2 + 2x + 4y  4 = 0. Ifthen a + b is equal to:          (JEE Main 2021)
(a) 3
(b) 11
(c) 5
(d) 7

Ans. c

Centre of smallest circle is A
Centre of largest circle is B
r= |CP − CA| = 3√2 − 3
r1 = CP − CB = 3√2 + 3

a = 3, b = 2

Q.39. Let S1 : x2 + y2 = 9 and S2 : (x  2)2 + y2 = 1. Then the locus of center of a variable circle S which touches S1 internally and S2 externally always passes through the points:          (JEE Main 2021)
(a)
(b) (1, ± 2)
(c) (2, ±3/2)
(d) (0, ±√3)

Ans. c
S1 : x2 + y2 = 9 ; C1 (0, 0), r1 = 3
S2 : (x  2)2 + y2 = 1 ; C2 (2, 0), r2 = 1
Image
Let the variable circle S and its radius is r units.
Here S and S1 touches internally
Distance between center,
S + S1 = PC1 = 3  r
Here S and S2 touches externally
Distance between center,
S + S2 = PC2 = 1 + r
PC1 + PC2 = 4 > C1 C2
So locus is ellipse whose focii are C1 & C2 and major axis is 2a = 4 and 2ae = C1C2 = 2
e = 1/2

Centre of ellipse is midpoint of C1 & C2 is (1, 0)
Equation of ellipse is
Now by cross checking the option (2, ±3/2) satisfied it.

Q.40. Let A = {(x, y) ∈ R × R|2x2 + 2y2 − 2x − 2y = 1}
B = {(x, y) ∈ R × R|4x2 + 4y2 − 16y + 7 = 0}
and
C = {(x, y) ∈ R × R|x+ y2 − 4x − 2y + 5 ≤ r2}.
Then the minimum value of |r| such that A ∪ B ⊆ C is equal to          (JEE Main 2021)
(a)
(b)
(c)
(d) 1 + √5

Ans. c

S3 = x2 + y2 − 4x − 2y + 5 − r2 = 0
C3 (2, 1)

Q.41. Two tangents are drawn from the point P(1, 1) to the circle x2 + y2  2x  6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to:           (JEE Main 2021)
(a) 2
(b)
(c) 4
(d)

Ans. c

Q.42. For the four circles M, N, O and P, following four equations are given:
Circle M : x2 + y2 = 1
Circle N : x2 + y2  2x = 0
Circle O : x2 + y2  2x  2y + 1 = 0
Circle P : x2 + y2  2y = 0
If the centre of circle M is joined with centre of the circle N, further center of circle N is joined with centre of the circle O, centre of circle O is joined with the centre of circle P and lastly, centre of circle P is joined with centre of circle M, then these lines form the sides of a:            (JEE Main 2021)
(a) Rhombus
(b) Square
(c) Rectangle
(d) Parallelogram

Ans. b
CM = (0, 0)
CN = (1, 0)
CO = (1, 1)
CP = (0, 1)

Q.43. Choose the correct statement about two circles whose equations are given below:
x2 + y2  10x  10y + 41 = 0
x2 + y2  22x  10y + 137 = 0             (JEE Main 2021)
(a) circles have same centre
(b) circles have no meeting point
(c) circles have only one meeting point
(d) circles have two meeting points

Ans. c
Let S1: x2 + y2  10x  10y + 41 = 0
⇒ (x − 5)2 + (y − 5)2 = 9
Centre (C1) = (5, 5)
S2: x2 + y2  22x  10y + 137 = 0
⇒ (x − 11)2 + (y − 5)2 = 9
Centre (C2) = (11, 5)
distance (C1C2) =
distance (C1C2) = 6
r+ r= 3 + 3 = 6
circles touch externally
Hence, circle have only one meeting point.

Q.44. Two tangents are drawn from a point P to the circle x2 + y2  2x  4y + 4 = 0, such that the angle between these tangents itan−1(12/5), where tan−1(12/5)  (0π). If the centre of the circle is denoted by C and these tangents touch the circle at points A and B, then the ratio of the areas of ΔPAB and ΔCAB is:              (JEE Main 2021)
(a) 3 : 1
(b) 9 : 4
(c) 2 : 1
(d) 11 : 4

Ans. b

In ΔCAP,

In ΔAPM,

In ΔCAM,

Q.45. Let the tangent to the circle x2 + y2 = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r2 is equal to:              (JEE Main 2021)
(a) 585/66
(b) 625/72
(c) 529/64
(d) 125/72

Ans. b
Given equation of circle
x2 + y2 = 25
Tangent equation at (3, 4)
T : 3x + 4y = 25

Incentre of ΔOPQ.

Distance from origin to incenter is r.

Therefore, the correct answer is (b).

Q.46. Choose the incorrect statement about the two circles whose equations are given below:
x2 + y2  10x  10y + 41 = 0 and
x2 + y2  16x  10y + 80 = 0               (JEE Main 2021)
(a) Distance between two centres is the average of radii of both the circles.
(b) Both circles pass through the centre of each other.
(c) Circles have two intersection points.
(d) Both circle's centers lie inside region of one another.

Ans. d
S1  x2 + y2  10x  10y + 41 = 0
Centre C1  (5, 5), radius r1 = 3
S2  x2 + y2  16x  10y + 80 = 0
Centre C2  (8, 5), radius r2 = 3
Distance between centres = 3
Hence both circles pass through the centre of each other, have two intersection point and distance between two centres in average of radii of both the circles.
Hence, option (d) is the incorrect statement.

Q.47. The line 2x − y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x − 2y = 4. Then, the radius of the circle is:               (JEE Main 2021)
(a) 5√3
(b) 4√5
(c) 3√5
(d) 5√4

Ans. c

m1 × m2 = −1
a − 14 = 2 − a
2a = 16
a = 8
∴ Centre (8, 2)
= √45
= 3√5

Q.48. Let the lengths of intercepts on x-axis and y-axis made by the circle x2 + y2 + ax + 2ay + c = 0, (a < 0) be 2√2 and 2√5, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line x + 2y = 0, is equal to:                (JEE Main 2021)
(a) 10
(b) 6
(c) 11
(d) 7

Ans. b

a2 − 4c = 8 .... (1)

a− c = 5 .... (2)
(2) − (1)
3c = −3a ⇒ c = −1
a2 = 4 ⇒ a = −2 (Given a < 0)
Equation of circle
x2 + y2 − 2x − 4y − 1 = 0
Equation of tangent which is perpendicular to the line x + 2y = 0 is
2x − y + λ = 0
∴ p = r

⇒ λ = ±√30
∴ Tangent 2x − y ± √30 = 0
Distance from origin

Q.49. If the locus of the mid-point of the line segment from the point (3, 2) to a point on the circle, x2 + y2 = 1 is a circle of radius r, then r is equal to:                 (JEE Main 2021)
(a) 1/4
(b) 1/2
(c) 1
(d) 1/3

Ans. b
Let P(h, k) and point on the circle is (cosθ, sinθ)

cosθ = 2h − 3 and sinθ = 2h − 2
(2h − 3)2 + (2h − 2)2 = 1
⇒ 4x2 − 12x + 9 + 4y2 − 8y + 4 = 1
⇒ 4x+ 4y2 − 12x − 8y + 12 = 0
⇒ x2 + y− 3x − 2y + 3 = 0

Q.50. Let A(1, 4) and B(1, 5) be two points. Let P be a point on the circle (x  1)2 + (y  1)2 = 1 such that (PA)2 + (PB)2 have maximum value, then the points, P, A and B lie on:                  (JEE Main 2021)
(a) a straight line
(b) an ellipse
(c) a parabola
(d) a hyperbola

Ans. a
P be a point on (x − 1)2 + (y − 1)= 1
so P(1 + cos⁡θ, 1 + sin⁡θ)
A(1, 4), B(1, 5)
(PA)2 + (PB)2
= (cos⁡θ)2 + (sin⁡θ − 3)2 + (cos⁡θ)2 + (sin⁡θ + 6)2
= 47 + 6sin⁡θ
It is maximum if sin⁡θ = 1
When sin⁡θ = 1, cos⁡θ = 0
So P(1, 2), A(1, 4), B(1, 5)
P, A, B are collinear points.

Q.51. In the circle given below, let OA = 1 unit, OB = 13 unit and PQ  OB. Then, the area of the triangle PQB (in square units) is:                  (JEE Main 2021)
(a) 24√2
(b) 24√3
(c) 26√2
(d) 26√3

Ans. b

Let PA = AQ = λ
OA . AB = AP . AQ
⇒ 1.12 = λ . λ
⇒ λ = 2√3

= 2√4

Q.52. If the curve x2 + 2y2 = 2 intersects the line x + y = 1 at two points P and Q, then the angle subtended by the line segment PQ at the origin is:                  (JEE Main 2021)
(a)
(b)
(c)
(d)

Ans. d

Line : x + y = 1

Using homogenisation
x2 + 2y2 = 2(1)2
x2 + 2y= 2(x + y)2
x2 + 2y2 = 2x2 + 2y2 + 4xy
x2 + 4xy = 0
for ax2 + 2hxy + by2 = 0

tan⁡θ = −4

Q.53. Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices (a, c), (2, b) and (a, b) be (10/3, 7/3). If α, β are the roots of the equation ax2 + bx + 1 = 0, then the value of α2 + β2 − αβ is:                  (JEE Main 2021)
(a) 69/256
(b) 71/256
(c)
(d)

Ans. c
2b = a + c

a = 4,

b = 11/4
c = 3/2
The value of
α+ β2 − αβ
α2 + β2 + 2αβ − 3αβ
(α + β)2 − 3αβ

Q.54. Let B be the centre of the circle x2 + y2  2x + 4y + 1 = 0. Let the tangents at two points P and Q on the circle intersect at the point A(3, 1). Then 8.is equal to _____________.                  (JEE Main 2021)

Ans. 18

In ΔABP

AP2 = AB2 − BP2 = 13 − 4 = 9

AP = 3

AQ = AP = 3

Let ∠ABP = θ, ∠BAP = 90 − θ

In ΔABP, tanθ = 3/2

In ΔARP,

In ΔBRP,

cos⁡θ = BR/BP

Q.55. If the variable line 3x + 4y = α lies between the two circles (x  1)2 + (y  1)2 = 1 and (x  9)2 + (y  1)2 = 4, without intercepting a chord on either circle, then the sum of all the integral values of α is ___________.                   (JEE Main 2021)

Ans. 165

Both centers should lie on either side of the line as well as line can be tangent to circle.
(3 + 4  α) . (27 + 4  α) < 0
(7  α) . (31  α) < 0  α  (7, 31) ....... (1)
d1 = distance of (1, 1) from line
d2 = distance of (9, 1) from line

(1)  (2)  (3)  α  [12, 21]
Sum of integers = 165

Q.56. Two circles each of radius 5 units touch each other at the point (1, 2). If the equation of their common tangent is 4x + 3y = 10, and C1(α, β) and C2(γδ), C1  C2 are their centres, then |(α + β) (γ + δ)| is equal to ___________.                   (JEE Main 2021)

Ans. 40
Slope of line joining centres of circles = 4/3 = tan⁡θ

⇒ cos⁡θ = 3/5, sin⁡θ = 4/5
Now using parametric form

(x, y) = (1 + 5cosθ, 2 + 5sinθ)
(αβ) = (4, 6)
(x, y) = (γδ) = (1  5cosθ, 2  5sinθ)
(γ, s) = (2, 2)
|(α + β) (γ + δ)| = | 10x  4 | = 40

Q.57. Let the equation x2 + y2 + px + (1  p)y + 5 = 0 represent circles of varying radius r  (0, 5]. Then the number of elements in the set S = {q : q = p2 and q is an integer} is __________.                   (JEE Main 2021)

Ans. 61

Since, r ∈ (0, 5]
So, 0 < 2p2 − 2p − 19 ≤ 100

so, number of integral values of p2 is 61.

Q.58. The locus of a point, which moves such that the sum of squares of its distances from the points (0, 0), (1, 0), (0, 1), (1, 1) is 18 units, is a circle of diameter d. Then d2 is equal to _____________.                   (JEE Main 2021)

Ans. 16
Let point P(x, y)
A(0, 0), B(1, 0), C(0, 1), D(1, 1)
(PA)2 + (PB)2 + (PC)2 + (PD)2 = 18
x + y + x + (y − 1) + (x − 1) + y + (x − 1) + (y − 1)= 18
⇒ 4(x + y) − 4y − 4x = 14

⇒ d= 16

Q.59. The minimum distance between any two points P1 and P2 while considering point P1 on one circle and point P2 on the other circle for the given circles' equations
x2 + y