Let S 1 be a circle passing through A 0 , 1 and B - 2 , 2 and S 2 be a circle of radius 10 units such that A B is the common chord of S 1 and S 2 . Find the equation of S 2 .

S 1 is passing through A 0 , 1 and B - 2 , 2 Equation of line A B is y - 1 = 2 - 1 - 2 - 0 x - 0 ⇒ x + 2 y - 2 = 0 Circle S 1 whose diameter is A B is x - 0 x + 2 + y - 1 y - 2 = 0 ⇒ x 2 + y 2 + 2 x - 3 y + 2 = 0 . Equation of S 2 is x 2 + y 2 + 2 x - 3 y + 2 + λ x + 2 y - 2 = 0 Radius = λ + 2 2 2 + 2 λ - 3 2 2 - 2 - 2 λ = 10 given ⇒ λ 2 + 4 λ + 4 + 4 λ 2 + 9 - 12 λ - 4 2 - 2 λ = 10 × 4 ⇒ 5 λ 2 = 35 ⇒ λ = ± 7 Hence, S 2 is x 2 + y 2 + 2 x - 3 y + 2 ± 7 x + 2 y - 2 = 0 .

Let S 1 be a circle passing through A 0 , 1 and B - 2 , 2 and S 2 be a circle of radius 10 units such that A B is the common chord of S 1 and S 2 . Find the equation of S 2 .

S 1 is passing through A 0 , 1 and B - 2 , 2 Equation of line A B is y - 1 = 2 - 1 - 2 - 0 x - 0 ⇒ x + 2 y - 2 = 0 Circle S 1 whose diameter is A B is x - 0 x + 2 + y - 1 y - 2 = 0 ⇒ x 2 + y 2 + 2 x - 3 y + 2 = 0 . Equation of S 2 is x 2 + y 2 + 2 x - 3 y + 2 + λ x + 2 y - 2 = 0 Radius = λ + 2 2 2 + 2 λ - 3 2 2 - 2 - 2 λ = 10 given ⇒ λ 2 + 4 λ + 4 + 4 λ 2 + 9 - 12 λ - 4 2 - 2 λ = 10 × 4 ⇒ 5 λ 2 = 35 ⇒ λ = ± 7 Hence, S 2 is x 2 + y 2 + 2 x - 3 y + 2 ± 7 x + 2 y - 2 = 0 .

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 tangents 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?

The point 0 , 3 2 does NOT lie in E 1

E M B I B E

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 22

HARD

JEE Main/Advance

IMPORTANT

Earn 100

Let $S_{1}$ be a circle passing through $A (0, 1)$ and $B (- 2, 2)$ and $S_{2}$ be a circle of radius $\sqrt{10}$ units such that $A B$ is the common chord of $S_{1}$ and $S_{2}$ . Find the equation of $S_{2}$ .

Important Questions on Circle

MEDIUM

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 23

The curves whose equations are
$S = a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$

$S^{'} = a^{'} x^{2} + 2 h^{'} x y + b^{'} y^{2} + 2 g^{'} x + 2 f^{'} y + c^{'} = 0$ intersect in four concyclic points then find relation in $a, b, h, a^{'}, b^{'}, h^{'} .$

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 24

A circle of constant radius

' r'

passes through the origin

O

and cuts the axes of coordinates in points

P

and

Q

, then find the equation of the locus of the foot of the perpendicular from

O

to

P Q

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 25

The ends

A, B

of a fixed straight line of length

' a'

and ends

A^{'}

and

B^{'}

of another fixed straight line of length

' b'

slide upon the

x

-axis and

y

-axis (one end on axis of

x

and the other on axis of

y

). Find the locus of the centre of the circle passing through

A, B, A^{'}

and

B^{'}

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 26

A parabola is drawn to pass through

A

and

B,

the ends of a diameter of a given circle of radius

a,

and to have as directrix a tangent to a concentric circle of radius

b;

then axes being

A B

and a perpendicular diameter, prove that the locus of the focus of the parabola is

\frac{x^{2}}{b^{2}} + \frac{y^{2}}{b^{2} - a^{2}} = 1

.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 27

Find an equation of the hyperbola whose directrix is the normal to circle

x^{2} + y^{2} - 4 x - 6 y + 9 = 0

having slope is

2

and eccentricity is equal to radius of given circle when focus of hyperbola is centre of the given circle.

HARD

JEE Main/Advance

IMPORTANT

Alpha Question Bank for Engineering: Mathematics>Chapter 16 - Circle>Exercise-4>Q 28

Let $P$ be a point from where perpendicular tangents are drawn to the circle $2 x^{2} + 2 y^{2} - a^{2} = 0$ . Let a line from $P$ perpendicular to $O P$ is drawn which intersect hyperbola $x^{2} - y^{2} = a^{2}$ at $Q$ and $R$ . Find number of all possible positions of $P$ such that product of ordinates of points $Q$ and $R$ is $\frac{3}{2} a^{2}$ .