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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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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)
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Radius of circumcircle (R) = √10/2

∴ Center of circle (M) =JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

Now distance between center of two circles,

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

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,

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

Option A:

When n = 4,
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Option (A) is wrong.

Option B:

When n = 5,

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

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

∴ cosec45∘ < cosec36 < cosec30

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

∴ R < r, when n = 5

∴ Option (B) is wrong.

Option C:

When n = 8,

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

cosecθ is decreasing function in 0 to π/2.

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

∴ Option (C) is correct.

Option D:

When n = 12, then

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
sin⁡θ = 12/13
In △PAO:
PO/OA = sec⁡θ
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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

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

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

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

Then (−g, −f) = (2, 2)

∴ g = f = −2 .......... (i)

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


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 centreJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 These two circles touch internally hence 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ (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 =JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis 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.JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

r from centre to lineJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis radius of circle
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
So (5h − 8K)2 + 5r2

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

= 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

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

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

∵ P lies of L1

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

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

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


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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
As slope of line joining (1, 2) and (3, 6) is 2 given diameter is parallel to side 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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

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

⇒ 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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Here, OM2 = OP2 − PM2
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

Ans. b
x2 + y2 − 4x + 3 = 0

⇒ (x − 2)2 + y2 = 1
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Also, AO = BO

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


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

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

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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
QR = 2r = 4

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

= 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 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Then
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(orthocentre coincide with centroid)

∴ (3h − a)2 + (3k − b)2 = 2

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

∵ 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

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


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

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

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

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

Distance of y = mx + c line from center (2, 1) is,
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
From (1) and (2), we get

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

⇒ m − 1 + c = ±c ..... (4)

taking positive sign,

m − 1 + c = c

⇒ m − 1 = 0

⇒ m = 1

From (2) and (3), we get

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

⇒ (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

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

⇒ c = 1/4

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

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

∴ 20(r2 + c)

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

Given,

equation of circle is

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

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

= 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

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

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

Now area of the triangle ANB

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


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

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


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,JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE) 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,JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ) lies on or inside the C
 JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, circle lies in 4th quadrant centre ≡ (3/2, −1)

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


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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Solving it with equation of given circle, 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
But x ≠ 5/2 because AB is not the diameter.

So, centre will be (15/2, 1/2)
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 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)

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

⇒ 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 thatJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEIf 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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
a + b = 7λ

b + c = 8λ

c + a = 9λ

a + b + c = 12λ

∴ a = 4λ, b = 3λ, c = 5λ
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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)

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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.
The radius of the circle C is ___________.        (JEE Advanced 2021)

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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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 JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE, 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 withJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEThe number of all those circles Dn that are inside M is
       (JEE Advanced 2021)
(a) 198
(b) 199
(c) 200
(d) 201

Ans. b
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 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 JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE, 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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 
For circle Cn to be inside M. 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 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).
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 
Then, 1  2g = 0  g = 1 / 2
and 5 + 1  4f = 0  f = 3 / 2
 Equation of circumcircle isJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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

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


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

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


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)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b) 100 < C < 165
(c) 81 < C < 156
(d) 100 < C < 156

Ans. d
S : 36x2 + 36y2  108x + 120y + C = 0 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Now, 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 
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. IfJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen a + b is equal to:          (JEE Main 2021) 
(a) 3
(b) 11
(c) 5
(d) 7

Ans. c
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Centre of smallest circle is A
Centre of largest circle is B
r= |CP − CA| = 3√2 − 3
r1 = CP − CB = 3√2 + 3
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Centre of ellipse is midpoint of C1 & C2 is (1, 0)
Equation of ellipse isJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d) 1 + √5

Ans. c
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
S3 = x2 + y2 − 4x − 2y + 5 − r2 = 0
C3 (2, 1) 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c) 4
(d)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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


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)

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


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)
Radius r1 = 3
S2: x2 + y2  22x  10y + 137 = 0
⇒ (x − 11)2 + (y − 5)2 = 9
Centre (C2) = (11, 5)
Radius r2 = 3
distance (C1C2) =JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔCAP, 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔAPM,
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔCAM, 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Incentre of ΔOPQ. 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 Distance from origin to incenter is r.
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
m1 × m2 = −1
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEa − 14 = 2 − a
2a = 16
a = 8
∴ Centre (8, 2)
Radius =JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= √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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
a2 − 4c = 8 .... (1)
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ λ = ±√30
∴ Tangent 2x − y ± √30 = 0
Distance from originJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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)
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
cosθ = 2h − 3 and sinθ = 2h − 2
Squaring and adding we get
(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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 


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)
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(a) 24√2
(b) 24√3
(c) 26√2
(d) 26√3

Ans. b
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Let PA = AQ = λ
OA . AB = AP . AQ
⇒ 1.12 = λ . λ
⇒ λ = 2√3
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
= 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)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(b)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(c)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(d)JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. d
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Line : x + y = 1
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
tan⁡θ = −4
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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

Ans. c
2b = a + c 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
a = 4, 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
b = 11/4
c = 3/2
 Quadratic Equation isJEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE 
 The value of
α+ β2 − αβ
α2 + β2 + 2αβ − 3αβ
(α + β)2 − 3αβ
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


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.JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis equal to _____________.                  (JEE Main 2021) 

Ans. 18
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔABP

AP2 = AB2 − BP2 = 13 − 4 = 9

AP = 3

AQ = AP = 3

Let ∠ABP = θ, ∠BAP = 90 − θ

In ΔABP, tanθ = 3/2
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔARP, 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔBRP,

cos⁡θ = BR/BP

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


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

JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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 
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ cos⁡θ = 3/5, sin⁡θ = 4/5
Now using parametric form
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Since, r ∈ (0, 5]
So, 0 < 2p2 − 2p − 19 ≤ 100
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
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
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
JEE Main Previous year questions (2021-22): Circle - Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
⇒ 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 + y2  10x  10y + 41 = 0
x2 + y2