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If the circle C₁ : x² + y² = 16 intersects another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/4, then the coordinates of the centre of C₂ are :

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Important Questions on Circle

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

The area (in sq. units) of the triangle formed by the two tangents drawn from the extremal point

O (0, 0)

to the circle

x^{2} + y^{2} - 2 g x - 2 h y + h^{2} = 0

and their chord of contact is

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

A line

y = m x + 1

intersects the circle

{(x - 3)}^{2} + {(y + 2)}^{2} = 25

at the points

P

and

Q .

If the midpoint of the line segment

P Q

has

x -

coordinate

- \frac{3}{5},

then which one of the following options is correct?

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

The angle between the pair of tangents drawn from

(1, 3)

to the circle

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

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

The angle between a pair of tangents drawn from a point

P

to the circle

x^{2} + y^{2} + 4 x - 6 y + 9 \sin^{2} α + 13 \cos^{2} α = 0

2 α

. The equation of the locus of the point

P

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

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}

P

and tangent to

S_{2}

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)

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?

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

The lines represented by

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

are normals to a circle

S = 0

. If this circle touches the circle

S^{'} \equiv x^{2} + y^{2} - 2 x + 2 y - 7 = 0

externally, then the equation of the chord of contact of centre of

S^{'} = 0

with respect to

S = 0

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

Let the circles

C_{1} : x^{2} + y^{2} = 9

and

C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16,

intersect at the points

X

and

Y .

Suppose that another circle

C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}

satisfies the following conditions:

(i)

centre of

C_{3}

is collinear with the centres of

C_{1}

and

C_{2}

(i i) C_{1}

and

C_{2}

both lie inside

C_{3},

and

(i i i) C_{3}

touches

C_{1}

M

and

C_{2}

N .

Let the line through

X

and

Y

intersect

C_{3}

Z

and

W,

and let a common tangent of

C_{1}

and

C_{3}

be a tangent to the parabola

x^{2} = 8 α y .

There are some expression given in the List-

I

whose values are given in List-

I I

below:

	List- $I$		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

P (x_{1}, y_{1})

is a point such that the lengths of the tangents from it to the circles

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

and

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

are in the ratio

2 : 3

, then the locus of

P

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

If the chord of contact of tangents from a point on the circle

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

to the circle

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

touches the circle

x^{2} + y^{2} = r_{3}^{2}

, then

r_{1}, r_{2}, r_{3}

are in:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

Let

C

be the circle with center at

(1, 1)

and radius

= 1

. If

T

is the circle centered at

(0, y)

, passing through the origin and touching the circle

C

externally, then the radius of

T

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

A chord

A B

is drawn from the point

A (0, 3)

on the circle

x^{2} + 4 x + (y - 3)^{2} = 0,

and is extended to

M

such that

A M = 2 A B

. The locus of

M

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

A line drawn through the point

P (4, 7)

cuts the circle

x^{2} + y^{2} = 9

at the points

A

and

B

. Then

P_{A} \cdot P_{B}

is equal to.

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

Let

A

be the centre of the circle

x^{2} + y^{2} - 2 x - 4 y - 20 = 0 .

Let

B (1, 7)

and

D (4, - 2)

be two points on the circle such that tangents at

B

and

D

meet at

C

. The area of the quadrilateral

A B C D

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

If the chord

L \equiv y - m x - 1 = 0

of the circle

S \equiv x^{2} + y^{2} - 1 = 0

touches the circle

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

, then the possible points for which

L = 0

is a chord of contact of

S = 0

are

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

Let

T

be a circle with diameter

A B

and centre

O

. Let

l

be the tangent to

T

B

. For each point

M

T

different from

A

, consider the tangent

t

M

and let interest

l

P .

Draw a line parallel to

A B

through

P

intersecting

O M

Q .

The locus of

Q

M

varies over

T

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

Two tangents are drawn from a point

P

to the circle

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

, such that the angle between these tangents is

\tan^{- 1} (\frac{12}{5})

, where

\tan^{- 1} (\frac{12}{5}) \in (0, π) .

If the centre of the circle is denoted by

C

and these tangents touch the circle at points

A

and

B

, then the ratio of the areas of

Δ P A B

and

Δ C A B

is :

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

In the circle given below, let

O A = 1

unit,

O B = 13

unit and

P Q ⊥ O B

. Then, the area of the triangle

P Q B

(in square units) is :

Question Image

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of 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 ___________.

MEDIUM

Mathematics>Coordinate Geometry>Circle>Common Chord of Two Circles

If the length of the tangent from any point on the circle

{(x - 3)}^{2} + {(y + 2)}^{2} = 5 r^{2}

to the circle

{(x - 3)}^{2} + {(y + 2)}^{2} = r^{2}

16

units, then the area between the two circles in sq. units is

HARD

Mathematics>Coordinate Geometry>Circle>Common Chord of 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

If the circle C1 : x2 + y2 = 16 intersects another circle C2 of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/4, then the coordinates of the centre of C2 are :

Important Questions on Circle

Learn and Prepare for any exam you want !

If the circle C₁ : x² + y² = 16 intersects another circle C₂ of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/4, then the coordinates of the centre of C₂ are :