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

The possible number of values of a for which the common chord of the circles x2y28 and xa2y28 subtends a right angle at the origin is

MEDIUM

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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:

MEDIUM

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

Three circles of radii

a, b, c, (a < b < c)

touch each other externally. If they have

x -

axis as a common tangent, then:

HARD

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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

is

HARD

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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

is

HARD

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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:

EASY

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

If the circles given by

S \equiv x^{2} + y^{2} - 14 x + 6 y + 33 = 0

and

S^{'} \equiv x^{2} + y^{2} - a^{2} = 0 (a \in N)

have

4

common tangents, then the possible number of circles

S^{'} = 0

is

MEDIUM

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

The number of common tangents that can be drawn to two given circles is at the most

HARD

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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

is

\frac{π}{3},

then a value of

k

is

HARD

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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

MEDIUM

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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

is

HARD

Mathematics>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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>Geometry - II>Circle>Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle

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:

The possible number of values of $a$ for which the common chord of the circles $x^{2} + y^{2} = 8$ and ${(x - a)}^{2} + y^{2} = 8$ subtends a right angle at the origin is

Important Questions on Circle

Learn and Prepare for any exam you want !

The possible number of values of a for which the common chord of the circles x2+y2=8 and x-a2+y2=8 subtends a right angle at the origin is

Important Questions on Circle

Learn and Prepare for any exam you want !

The possible number of values of $a$ for which the common chord of the circles $x^{2} + y^{2} = 8$ and ${(x - a)}^{2} + y^{2} = 8$ subtends a right angle at the origin is