The polars of - 1 , 2 with respect to the two circles S 1 ≡ x 2 + y 2 + 6 y + 7 = 0 and S 2 ≡ x 2 + y 2 + 6 x + 1 = 0 are

Intersecting at a non zero point

HARD

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Consider a circle S with centre at the origin and radius $4$ . Four circles $A, B, C$ and $D$ each with radius unity and centres $(- 3, 0), (- 1, 0), (1, 0)$ and $(3, 0)$ respectively are drawn. $A$ chord $P Q$ of the circle $S$ touches the circle $B$ and passes through the centre of the circle $C$ . If the length of this chord can be expresses as $\sqrt{x}$ , find $x$ .

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Important Questions on Circle

HARD

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

The angle between the pair of tangents drawn from

(1, 3)

to the circle

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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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

HARD

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

If the lines

2 x + y + 12 = 0, k x - 3 y - 10 = 0

are conjugate with respect to the circle

x^{2} + y^{2} - 4 x + 3 y - 1 = 0,

then

k =

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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

is a circle to which tangents are drawn from the point

(4, 5)

which form a quadrilateral with a pair of radii. The area of this quadrilateral in square units is ___________.

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

Consider the circles

S_{1} : x^{2} + y^{2} + 2 x + 8 y - 23 = 0

and

S_{2} : x^{2} + y^{2} - 4 x + 10 y + 19 = 0

If the polars of the centre of a circle with respect to the another circle are

L_{1}

and

L_{2}

, then

L_{1}, L_{2}

are

HARD

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

The area of the quadrilateral formed by the tangents from the point

(4, 7)

with a pair of radii of the circle

x^{2} + y^{2} - 4 x - 6 y - 3 = 0

EASY

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

The locus of the mid points of the chords of the circle

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

drawn from a point

(0, 0)

on it is

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

The polars of

(- 1, 2)

with respect to the two circles

S_{1} \equiv x^{2} + y^{2} + 6 y + 7 = 0

and

S_{2} \equiv x^{2} + y^{2} + 6 x + 1 = 0

are

MEDIUM

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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>Tangent and Normal of a Circle

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:

HARD

Mathematics>Coordinate Geometry>Circle>Tangent and Normal of a Circle

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.

Important Questions on Circle

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