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

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Let $G$ be a circle of radius $R > 0$ . Let $G_{1}, G_{2}, \dots, G_{n}$ be $n$ circles of equal radius $r > 0$ . Suppose each of the $n$ circles $G_{1}, G_{2}, \dots, G_{n}$ touches the circle $G$ externally. Also, for $i = 1, 2, \dots, n - 1$ , the circle $G_{i}$ touches $G_{i + 1}$ externally, and $G_{n}$ touches $G_{1}$ externally. Then, which of the following statements is/are TRUE?

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Important Questions on Circle

HARD

Mathematics>Coordinate Geometry>Circle>Interaction Between Two Circles

The two circles

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

and

x^{2} + y^{2} - 10 x + 16 = 0

intersect at two distinct points. Then which one of the following is correct?

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

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>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

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 ____________.

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction Between Two 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>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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction Between Two Circles

The locus of the centre of the circles which touch both the circles $x^{2} + y^{2} = a^{2}$ and $x^{2} + y^{2} = 4 a x$ externally is

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:

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 :

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction Between Two 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

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-

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction Between Two 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>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

HARD

Mathematics>Coordinate Geometry>Circle>Interaction Between Two 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

EASY

Mathematics>Coordinate Geometry>Circle>Interaction Between Two Circles

Choose the incorrect statement about the two circles whose equations are given below:

$x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 16 x - 10 y + 80 = 0$

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction Between Two Circles

If one of the diameters of the circle

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

is a chord of another circle

{}^{'}C^{'}

, whose center is at

(2, 1)

, then its radius 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

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:

Important Questions on Circle

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