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 2 3 and centres Q 2 and Q 3 , respectively. If Q 2 and Q 3 lie on the y - axis, then

area of the triangle O R 2 R 3 is 6 2

area of the triangle P Q 2 Q 3 is 4 2

Let n ≥ 3 and let C 1 , C 2 , … … , C n , be circles with radii r 1 , r 2 , … . . , r n ,respectively. Assume that C i & C i + 1 touch externally for 1 ≤ i ≤ n - 1 . It is also given that the x - axis and the line y = 2 2 x + 10 are tangent to each of the circles. Then r 1 , r 2 , … . . , r n are in-

A geometric progression with common ratio 2 + 3

An arithmetic progression with common difference 3 + 2

A geometric progression with common ratio 3 + 2

An arithmetic progression with common difference 2 + 3

HARD

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If the circles $x^{2} + y^{2} + (3 + \sin β) x + (2 \cos α) y = 0$ and $x^{2} + y^{2} + (2 \cos α) x + 2 c y = 0$ touch each other, then the maximum value of $c$ is lesser than or equal to

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Important Questions on Circle

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Let the point

B

be the reflection of the point

A (2, 3)

with respect to the line

8 x - 6 y - 23 = 0 .

Let

T_{A}

and

T_{B}

be circle of radii

2

and

1

with centres

A

and

B

respectively. Let

T

be a common tangent to the circle

T_{A}

and

T_{B}

such that both the circle are on the same side of

T .

C

is the point of intersection of

T

and the line passing through

A

and

B,

then the length of the line segment

A C

is ___________.

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The number of common tangents to the circles

x^{2} + y^{2} - 4 x - 6 y - 12 = 0

and

x^{2} + y^{2} + 6 x + 18 y + 26 = 0

, is

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The common tangent to the circles

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

and

x^{2} + y^{2} + 6 x + 8 y - 24 = 0

also passes through the point:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The value of

λ

, for which the circle

x^{2} + y^{2} + 2 λ x + 6 y + 1 = 0

intersects the circle

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

orthogonally, is

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

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}

R_{2}

and

R_{3}

, respectively. Suppose

C_{2}

and

C_{3}

have equal radii

2 \sqrt{3}

and centres

Q_{2}

and

Q_{3}

, respectively. If

Q_{2}

and

Q_{3}

lie on the y - axis, then

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The locus of the centres of the circles, which touch the circle,

x^{2} + y^{2} = 1

externally, also touch the

y

-axis and lie in the first quadrant, is:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the curves,

x^{2} - 6 x + y^{2} + 8 = 0

and

x^{2} - 8 y + y^{2} + 16 - k = 0, (k > 0)

touch each other at a point, then the largest value of

k

is ____________.

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the angle between the circles

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

and

x^{2} + y^{2} + 8 x - 4 y + 11 = 0

\frac{π}{3},

then a value of

k

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The number of common tangents to the two circles

x^{2} + y^{2} - 8 x + 2 y = 0

and

x^{2} + y^{2} - 2 x - 16 y + 25 = 0

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

A circle S passes through the point

(0, 1)

and is orthogonal to the circles

{(x - 1)}^{2} + y^{2} = 16

and

x^{2} + y^{2} = 1

. Then

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

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 & Q

, then the line

4 x + 5 y - K = 0

, passes through

P

and

Q,

for:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the circles

x^{2} + y^{2} - 16 x - 20 y + 164 = r^{2}

and

{(x - 4)}^{2} + {(y - 7)}^{2} = 36

intersect at two distinct points, then:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Let the latus rectum of the parabola $y^{2} = 4 x$ be the common chord to the circles $C_{1}$ and $C_{2}$ each of them having radius $2 \sqrt{5}$ . Then, the distance between the centres of the circles $C_{1}$ and $C_{2}$ is :

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the angle of intersection at a point where the two circles with radii

5 c m

and

12 c m

intersect is

90 °

, then the length (in cm) of their common chord is:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the circle

S \equiv x^{2} + y^{2} - 4 = 0

intersects another circle

S^{'} = 0

of radius

\frac{5 \sqrt{2}}{2}

in such a manner that the common chord is of maximum length with slope equal to

\frac{1}{4},

then the centre of

S^{'} = 0

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

The condition for the circles

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

and

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

to touch each other is

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If the equation of the circle which passes through the point

(1, 1)

and cuts both the circles

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

and

x^{2} + y^{2} + 6 x - 4 y + 15 = 0

orthogonally is

x^{2} + y^{2} + 2 g x + 2 f y + c = 0,

then

5 g + 2 f + c =

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

If one of the diameters of the circle, given by the equation,

x^{2} + y^{2} - 4 x + 6 y - 12 = 0,

is a chord of a circle

S

, whose centre is at

(- 3, 2)

, then the radius of

S

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Let

n \geq 3

and let

C_{1}, C_{2}, \dots \dots, C_{n}

, be circles with radii

r_{1}, r_{2}, \dots . ., r_{n}

,respectively. Assume that

C_{i} & C_{i + 1}

touch externally for

1 \leq i \leq n - 1

. It is also given that the

x

- axis and the line

y = 2 \sqrt{2} x + 10

are tangent to each of the circles. Then

r_{1}, r_{2}, \dots . ., r_{n}

are in-

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Two Circles

Let

R

be the region of the disc

x^{2} + y^{2} \leq 1

in the first quadrant. Then, the area of the largest possible circle contained in

R

If the circles x2+y2+(3+sinβ)x+(2cosα)y=0 and x2+y2+(2cosα)x+2cy=0 touch each other, then the maximum value of c is lesser than or equal to

Important Questions on Circle

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If the circles $x^{2} + y^{2} + (3 + \sin β) x + (2 \cos α) y = 0$ and $x^{2} + y^{2} + (2 \cos α) x + 2 c y = 0$ touch each other, then the maximum value of $c$ is lesser than or equal to