Core Syllabus Module

Circles

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2016

LEVELJEE Advanced

The circle $C_{1} : x^{2} + y^{2} = 3$ , with centre at $O$ , intersects the parabola $x^{2} = 2 y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_{1}$ at $P$ touches other two circles $C_{2}$ and $C_{3}$ at $R_{2}$ and $R_{3}$ , respectively. Suppose $C_{2}$ and $C_{3}$ have equal radii $23$ and centres $Q_{2}$ and $Q_{3}$ , respectively. If $Q_{2}$ and $Q_{3}$ lie on the $y$ -axis, then

* Multiple Correct Options

(A)

Q_{2} Q_{3} = 12

(B)

R_{2} R_{3} = 46

(C)

area of the triangle

Δ O R_{2} R_{3}

62

(D)

area of the triangle

Δ P Q_{2} Q_{3}

42

JEE Advanced 1990

LEVELJEE Advanced

A circle touches the line $y = x$ at a point $P$ such that $O P = 42$ , where $O$ is the origin. The circle contains the point $(- 10, 2)$ in its interior and the length of its chord on the line $x + y = 0$ is $62$ . Determine the equation of the circle.

JEE Advanced 2018

LEVELJEE Advanced

Let $T$ be the line passing through the points $P (- 2, 7)$ and $Q (2, - 5)$ . Let $F_{1}$ be the set of all pairs of circles $(S_{1}, S_{2})$ such that $T$ is tangent to $S_{1}$ at $P$ and tangent to $S_{2}$ at $Q$ , and also such that $S_{1}$ and $S_{2}$ touch each other at a point, say, $M$ . Let $E_{1}$ be the set representing the locus of $M$ as the pair $(S_{1}, S_{2})$ varies in $F_{1}$ . Let the set of all straight line segments joining a pair of distinct points of $E_{1}$ and passing through the point $R (1, 1)$ be $F_{2}$ . Let $E_{2}$ be the set of the mid-points of the line segments in the set $F_{2}$ . Then, which of the following statement(s) is (are) TRUE ?

* Multiple Correct Options

(A)

The point

(- 2, 7)

lies in

E_{1}

(B)

The point

(\frac{4}{5}, \frac{7}{5})

does NOT lie in

E_{2}

(C)

The point

(\frac{1}{2}, \frac{1}{2})

lies in

E_{2}

(D)

The point

(0, \frac{3}{2})

does NOT lie in

E_{1}

JEE Advanced 1999

LEVELJEE Advanced

Let $T_{1}, T_{2}$ be two tangents drawn from $(- 2, 0)$ onto the circle $C : x^{2} + y^{2} = 1$ . Determine the circles touching $C$ and $T_{1}, T_{2}$ as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.

JEE Advanced 2024

LEVELJEE Advanced

Let the straight line $y = 2 x$ touch a circle with centre $(0, α)$ , $α > 0$ , and radius $r$ at a point $A_{1}$ . Let $B_{1}$ be the point on the circle such the line segment $A_{1} B_{1}$ is a diameter of the circle. Let $α + r = 5 + 5$ . Match each entry in List-I to the correct entries in List-II.

List-I

(P)

α

equals

(Q)

r

equals

(R)

A_{1}

equals

(S)

B_{1}

equals

List-II

(1)

(- 2, 4)

(2)

5

(3)

(- 2, 6)

(4)

5

(5)

(2, 4)

JEE Advanced 2023

LEVELJEE Advanced

Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius $r$ with center at the point $A = (4, 1)$ , where $1 < r < 3$ . Two distinct common tangents $P Q$ and $S T$ of $C_{1}$ and $C_{2}$ are drawn. The tangent $P Q$ touches $C_{1}$ at $P$ and $C_{2}$ at $Q$ . The tangent $S T$ touches $C_{1}$ at $S$ and $C_{2}$ at $T$ . Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the x-axis at a point $B$ . If $A B = 5$ , then the value of $r^{2}$ is

JEE Advanced 2022

LEVELJEE Advanced

Let $A B C$ be the triangle with $A B = 1, A C = 3$ and $∠ B A C = \frac{π}{2}$ . If a circle of radius $r > 0$ touches the sides $A B, A C$ and also touches internally the circumcircle of the triangle $A B C$ , then the value of $r$ is _____________.

JEE Advanced 2021

LEVELJEE Advanced

Comprehension Passage

Let

M = {(x, y) \in R \times R : x^{2} + y^{2} \leq r^{2}}

, where

r > 0

. Consider the geometric progression

a_{n} = \frac{1}{2 ^{n - 1}}, n = 1, 2, 3, \dots

. Let

S_{0} = 0

and, for

n \geq 1

, let

S_{n}

denote the sum of the first

n

terms of this progression. For

n \geq 1

, let

C_{n}

denote the circle with center

(S_{n - 1}, 0)

and radius

a_{n}

, and

D_{n}

denote the circle with center

(S_{n - 1}, S_{n - 1})

and radius

a_{n}

Question 1:

Consider $M$ with $r = \frac{1025}{513}$ . Let $k$ be the number of all those circles $C_{n}$ that are inside $M$ . Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

(A)

k + 2 l = 22

(B)

2 k + l = 26

(C)

2 k + 3 l = 34

(D)

3 k + 2 l = 40

Question 2:

Consider $M$ with $r = \frac{( 2 ^{199} - 1 ) 2}{2 ^{198}}$ . The number of all those circles $D_{n}$ that are inside $M$ is

Let $O$ be the centre of the circle $x^{2} + y^{2} = r^{2}$ , where $r > \frac{5}{2}$ . Suppose $P Q$ is a chord of this circle and the equation of the line passing through $P$ and $Q$ is $2 x + 4 y = 5$ . If the centre of the circumcircle of the triangle $O P Q$ lies on the line $x + 2 y = 4$ , then the value of $r$ is ____.

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

A circle

C

of radius 1 is inscribed in an equilateral triangle

P QR

. The points of contact of

C

with the sides

P Q, QR, R P

are

D, E, F

, respectively. The line

P Q

is given by the equation

3 x + y - 6 = 0

and the point

D

(\frac{3 3}{2}, \frac{3}{2})

. Further, it is given that the origin and the centre of

C

are on the same side of the line

P Q

Question 1:

The equation of circle $C$ is

(A)

(x - 23)^{2} + (y - 1)^{2} = 1

(B)

(x - 23)^{2} + (y + \frac{1}{2})^{2} = 1

(C)

(x - 3)^{2} + (y + 1)^{2} = 1

(D)

(x - 3)^{2} + (y - 1)^{2} = 1

Question 2:

Points $E$ and $F$ are given by

(A)

(\frac{3}{2}, \frac{3}{2}), (3, 0)

(B)

(\frac{3}{2}, \frac{1}{2}), (3, 0)

(C)

(\frac{3}{2}, \frac{3}{2}), (\frac{3}{2}, \frac{1}{2})

(D)

(\frac{3 3}{2}, \frac{3}{2}), (\frac{3}{2}, \frac{1}{2})

Question 3:

Equations of the sides $QR, R P$ are

(A)

y = \frac{2}{3} x + 1, y = - \frac{2}{3} x - 1

(B)

y = \frac{1}{3} x, y = 0

(C)

y = \frac{3}{2} x + 1, y = - \frac{3}{2} x - 1

(D)

y = 3 x, y = 0

LEVELJEE Advanced

Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$ . If $P S$ and $R Q$ intersect at a point $X$ on the circumference of the circle, then $2 r$ equals

(A)

P Q \cdot R S

(B)

(P Q + R S) /2

(C)

2 P Q \cdot R S / (P Q + R S)

(D)

(P Q^{2} + R S^{2}) /2

LEVELJEE Advanced

The centre of a circle passing through the points $(0, 0), (1, 0)$ and touching the circle $x^{2} + y^{2} = 9$ is

(A)

(\frac{3}{2}, \frac{1}{2})

(B)

(\frac{1}{2}, \frac{3}{2})

(C)

(\frac{1}{2}, \frac{1}{2})

(D)

(\frac{1}{2}, - 22)

JEE Advanced 2012

LEVELJEE Advanced

Comprehension Passage

A tangent

P T

is drawn to the circle

x^{2} + y^{2} = 4

at the point

P (3, 1)

. A straight line

L

, perpendicular to

P T

is a tangent to the circle

(x - 3)^{2} + y^{2} = 1

Question 1:

A possible equation of $L$ is

(A)

x - 3 y = 1

(B)

x + 3 y = 1

(C)

x - 3 y = - 1

(D)

x + 3 y = 5

Question 2:

A common tangent of the two circles is

(A)

x = 4

(B)

y = 2

(C)

x + 3 y = 4

(D)

x + 22 y = 6

JEE Main 2026 (23 January Shift 2)

LEVELJEE Advanced

If the points of intersection of the ellipses $x^{2} + 2 y^{2} - 6 x - 12 y + 23 = 0$ and $4 x^{2} + 2 y^{2} - 20 x - 12 y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $ab + 18 r^{2}$ is

JEE Main 2026 (28 January Shift 2)

LEVELJEE Advanced

Let the circle $x^{2} + y^{2} = 4$ intersect $x$ -axis at the points $A (a, 0), a > 0$ and $B (b, 0)$ . Let $P (2 cos α, 2 sin α)$ , $0 < α < \frac{π}{2}$ and $Q (2 cos β, 2 sin β)$ be two points such that $(α - β) = \frac{π}{2}$ . Then the point of intersection of $A Q$ and $B P$ lies on :

(A)

x^{2} + y^{2} - 4 y - 4 = 0

(B)

x^{2} + y^{2} - 4 x - 4 y = 0

(C)

x^{2} + y^{2} - 4 x - 4 = 0

(D)

x^{2} + y^{2} - 4 x - 4 y - 4 = 0

JEE Main 2026 (21 January Shift 2)

LEVELJEE Advanced

If $P$ is a point on the circle $x^{2} + y^{2} = 4$ , $Q$ is a point on the straight line $5 x + y + 2 = 0$ and $x - y + 1 = 0$ is the perpendicular bisector of $P Q$ , then 13 times the sum of abscissa of all such points $P$ is .........

JEE Main 2025 (January)

LEVELJEE Advanced

Let the parabola $y = x^{2} + p x - 3$ , meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(- 1, - 1)$ passes through the points P, Q and R, then the area of $△ P QR$ is:

Let a given line $L_{1}$ intersects the $x$ and $y$ axes at $P$ and $Q$ , respectively. Let another line $L_{2}$ , perpendicular to $L_{1}$ , cut the $x$ and $y$ axes at $R$ and $S$ , respectively. Show that the locus of the point of intersection of the lines $P S$ and $QR$ is a circle passing through the origin.

JEE Advanced 1988

LEVELJEE Advanced

If the circle $C_{1} : x^{2} + y^{2} = 16$ intersects another circle $C_{2}$ of radius 5 in such a manner that common chord is of maximum length and has a slope equal to $3/4$ , then the coordinates of the centre of $C_{2}$ are .........

JEE Main 2019 (9 April)

LEVELJEE Advanced

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^{2} = 4 x$ at the point $(1, 2)$ and the x-axis is :-

(A)

4 π (2 - 2)

(B)

8 π (3 - 22)

(C)

4 π (3 - 2)

(D)

8 π (2 - 2)

Micro-Topic Strategy Matrix

Standard and General Equation of a Circle

Intercepts Made by a Circle

Position of a Point with Respect to a Circle

Equation of Tangent and Normal

Length of Tangent and Chord of Contact

Director Circle and Family of Circles

Condition for Orthogonality

Chapter Elite-20 Challenge Zone

List-I

List-II

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider M with r=2198(2199−1)2​​. The number of all those circles Dn​ that are inside M is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of circle C is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Points E and F are given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Equations of the sides QR,RP are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The centre of a circle passing through the points (0,0),(1,0) and touching the circle x2+y2=9 is

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A possible equation of L is

A common tangent of the two circles is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the points of intersection of the ellipses x2+2y2−6x−12y+23=0 and 4x2+2y2−20x−12y+35=0 lie on a circle of radius r and centre (a,b), then the value of ab+18r2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area (in sq. units) of the smaller of the two circles that touch the parabola, y2=4x at the point (1,2) and the x-axis is :-

Consider $M$ with $r = \frac{( 2 ^{199} - 1 ) 2}{2 ^{198}}$ . The number of all those circles $D_{n}$ that are inside $M$ is

The equation of circle $C$ is

Points $E$ and $F$ are given by

Equations of the sides $QR, R P$ are

The centre of a circle passing through the points $(0, 0), (1, 0)$ and touching the circle $x^{2} + y^{2} = 9$ is

A possible equation of $L$ is

If the points of intersection of the ellipses $x^{2} + 2 y^{2} - 6 x - 12 y + 23 = 0$ and $4 x^{2} + 2 y^{2} - 20 x - 12 y + 35 = 0$ lie on a circle of radius $r$ and centre $(a, b)$ , then the value of $ab + 18 r^{2}$ is

The area (in sq. units) of the smaller of the two circles that touch the parabola, $y^{2} = 4 x$ at the point $(1, 2)$ and the x-axis is :-