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

EASY

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Consider a family of circles cutting the circles x² + y² + 2x – 4y – 4 = 0 and x² + y² – 4x + 4y + 4 = 0 orthogonally. Show that the chords in which the circle 2x² + 2y² – 5x – 6y – 2 = 0 cuts the member of the family are concurrent at a point. If the coordinates of this point are (a, b), then find the value of 25(a + b).

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Important Questions on Circle

HARD

Mathematics>Coordinate Geometry>Circle>Family of 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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Family of 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>Family of 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>Family of 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>Family of 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>Family of 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 =

HARD

Mathematics>Coordinate Geometry>Circle>Family of 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 :

MEDIUM

Mathematics>Coordinate Geometry>Circle>Family of Circles

If the angle between the circles

x^{2} + y^{2} - 12 x - 6 y + 41 = 0

and

x^{2} + y^{2} + k x + 6 y - 59 = 0

45^{°}

, then a value of

k

MEDIUM

Mathematics>Coordinate Geometry>Circle>Family of 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>Family of 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>Family of 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>Family of 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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Family of Circles

The equation of the tangent at the point

(0, 3)

on the circle which cuts the circles

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

and

x^{2} + y^{2} - 12 x + 2 y + 3 = 0

orthogonally is

HARD

Mathematics>Coordinate Geometry>Circle>Family of 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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Family of Circles

The center of the circle passing through the point

(1, 0)

and cutting the circles

x^{2} + y^{2} - 2 x + 4 y + 1 = 0

and

x^{2} + y^{2} + 6 x - 2 y + 1 = 0

orthogonally is

HARD

Mathematics>Coordinate Geometry>Circle>Family of Circles

Two circles each of radius

5

units touch each other at

(1, 2)

and

4 x + 3 y = 10

is their common tangent. The equation of that circle among the two given circles, such that some portion of it lies in every quadrant is

MEDIUM

Mathematics>Coordinate Geometry>Circle>Family of Circles

If the angle between the circles

x^{2} + y^{2} + 4 x - 5 = 0

and

x^{2} + y^{2} + 2 λ y - 4 = 0

\frac{π}{3}

then

λ =

HARD

Mathematics>Coordinate Geometry>Circle>Family of 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>Family of 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>Family of 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-

Important Questions on Circle

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