Director Circle and Family of Circles

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JEE Main 2009

If $P$ and $Q$ are the points of intersection of the circles $x^{2} + y^{2} + 3 x + 7 y + 2 p - 5 = 0$ and $x^{2} + y^{2} + 2 x + 2 y - p^{2} = 0$ then there is a circle passing through $P, Q$ and $(1, 1)$ for:

p

(B)

all except two values of

p

(C)

exactly one value of

p

(D)

all values of

p

JEE Main 2005

LEVELJEE Main

If the circles $x^{2} + y^{2} + 2 a x + cy + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5 x + b y - a = 0$ passes through $P$ and $Q$ for

(A)

exactly one value of

a

(B)

no value of

a

(C)

infinitely many values of

a

(D)

exactly two values of

a

JEE Advanced 2007

LEVELJEE Main

Tangents are drawn from the point $(17, 7)$ to the circle $x^{2} + y^{2} = 169$ . STATEMENT-1 : The tangents are mutually perpendicular. because STATEMENT-2 : The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^{2} + y^{2} = 338$ .

(A)

Statement-1 is True, statement-2 is True; statement-2 is a correct explanation for Statement-1.

(B)

Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1.

(C)

Statement-1 is True, Statement-2 is False

(D)

Statement-1 is False, Statement-2 is True.

LEVELJEE Main

The equation of the circle passing through $(1, 1)$ and the points of intersection of $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is

(A)

4 x^{2} + 4 y^{2} - 30 x - 10 y - 25 = 0

(B)

4 x^{2} + 4 y^{2} + 30 x - 13 y - 25 = 0

(C)

4 x^{2} + 4 y^{2} - 17 x - 10 y + 25 = 0

(D)

none of these

LEVELJEE Main

Two circles $x^{2} + y^{2} = 6$ and $x^{2} + y^{2} - 6 x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point $(1, 1)$ is

(A)

x^{2} + y^{2} - 6 x + 4 = 0

(B)

x^{2} + y^{2} - 3 x + 1 = 0

(C)

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

Consider a family of circles passing through two fixed points $A (3, 7)$ and $B (6, 5)$ . Show that the chords in which the circle $x^{2} + y^{2} - 4 x - 6 y - 3 = 0$ cuts the members of the family are concurrent at a point. Find the coordinate of this point.

JEE Advanced 1986

LEVELJEE Main

The equation of the line passing through the points of intersection of the circles $3 x^{2} + 3 y^{2} - 2 x + 12 y - 9 = 0$ and $x^{2} + y^{2} + 6 x + 2 y - 15 = 0$ is .........

JEE Main 2019 (10 April Shift 1)

LEVELJEE Main

If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0$ , $(K \in R)$ , intersect at the points P and Q, then the line $4 x + 5 y - K = 0$ passes through P and Q for :

(A)

exactly two values of K

(B)

exactly one value of K

(C)

no value of K

(D)

infinitely many values of K

JEE Main 2019 (10 April Shift 1)

LEVELJEE Main

The line $x = y$ touches a circle at the point $(1, 1)$ . If the circle also passes through the point $(1, - 3)$ , then its radius is :

(A)

32

(B)

(C)

22

(D)

JEE Main 2020 - 4 Sep (Evening)

LEVELJEE Main

The circle passing through the intersection of the circles, $x^{2} + y^{2} - 6 x = 0$ and $x^{2} + y^{2} - 4 y = 0$ having its centre on the line, $2 x - 3 y + 12 = 0$ , also passes through the point:

(A)

(- 3, 6)

(B)

(- 1, 3)

(C)

(- 3, 1)

(D)

(1, - 3)

JEE Main 2021 (26 August Shift 2)

LEVELJEE Advanced

A circle $C$ touches the line $x = 2 y$ at the point $(2, 1)$ and intersects the circle $C_{1} : x^{2} + y^{2} + 2 y - 5 = 0$ at two points $P$ and $Q$ such that $P Q$ is a diameter of $C_{1}$ . Then the diameter of $C$ is :

(A)

75

(B)

15

(C)

285

(D)

415

JEE Main 2022 (25 July Shift 1)

LEVELJEE Advanced

The sum of diameters of the circles that touch (i) the parabola $75 x^{2} = 64 (5 y - 3)$ at the point $(\frac{8}{5}, \frac{6}{5})$ and (ii) the y-axis, is equal to ______.

JEE Advanced 2020

LEVELJEE Advanced

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 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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If P and Q are the points of intersection of the circles x2+y2+3x+7y+2p−5=0 and x2+y2+2x+2y−p2=0 then there is a circle passing through P,Q and (1,1) for:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the circles x2+y2+2ax+cy+a=0 and x2+y2−3ax+dy−1=0 intersect in two distinct points P and Q then the line 5x+by−a=0 passes through P and Q for

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the circle passing through (1,1) and the points of intersection of x2+y2+13x−3y=0 and 2x2+2y2+4x−7y−25=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two circles x2+y2=6 and x2+y2−6x+8=0 are given. Then the equation of the circle through their points of intersection and the point (1,1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the line passing through the points of intersection of the circles 3x2+3y2−2x+12y−9=0 and x2+y2+6x+2y−15=0 is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the circles x2+y2+5Kx+2y+K=0 and 2(x2+y2)+2Kx+3y−1=0, (K∈R), intersect at the points P and Q, then the line 4x+5y−K=0 passes through P and Q for :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The line x=y touches a circle at the point (1,1). If the circle also passes through the point (1,−3), then its radius is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The circle passing through the intersection of the circles, x2+y2−6x=0 and x2+y2−4y=0 having its centre on the line, 2x−3y+12=0, also passes through the point:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of diameters of the circles that touch (i) the parabola 75x2=64(5y−3) at the point (58​,56​) and (ii) the y-axis, is equal to ______.

If $P$ and $Q$ are the points of intersection of the circles $x^{2} + y^{2} + 3 x + 7 y + 2 p - 5 = 0$ and $x^{2} + y^{2} + 2 x + 2 y - p^{2} = 0$ then there is a circle passing through $P, Q$ and $(1, 1)$ for:

If the circles $x^{2} + y^{2} + 2 a x + cy + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5 x + b y - a = 0$ passes through $P$ and $Q$ for

The equation of the circle passing through $(1, 1)$ and the points of intersection of $x^{2} + y^{2} + 13 x - 3 y = 0$ and $2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0$ is

Two circles $x^{2} + y^{2} = 6$ and $x^{2} + y^{2} - 6 x + 8 = 0$ are given. Then the equation of the circle through their points of intersection and the point $(1, 1)$ is

The equation of the line passing through the points of intersection of the circles $3 x^{2} + 3 y^{2} - 2 x + 12 y - 9 = 0$ and $x^{2} + y^{2} + 6 x + 2 y - 15 = 0$ is .........

If the circles $x^{2} + y^{2} + 5 K x + 2 y + K = 0$ and $2 (x^{2} + y^{2}) + 2 K x + 3 y - 1 = 0$ , $(K \in R)$ , intersect at the points P and Q, then the line $4 x + 5 y - K = 0$ passes through P and Q for :

The line $x = y$ touches a circle at the point $(1, 1)$ . If the circle also passes through the point $(1, - 3)$ , then its radius is :

The circle passing through the intersection of the circles, $x^{2} + y^{2} - 6 x = 0$ and $x^{2} + y^{2} - 4 y = 0$ having its centre on the line, $2 x - 3 y + 12 = 0$ , also passes through the point:

The sum of diameters of the circles that touch (i) the parabola $75 x^{2} = 64 (5 y - 3)$ at the point $(\frac{8}{5}, \frac{6}{5})$ and (ii) the y-axis, is equal to ______.