HARD

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Consider a circle in $x y$ -plane with diameter $1$ , passing through the origin $O$ and through the point A given as $(1, 0) .$ $B$ is any point on the circle. Let $C$ be the point of intersection of line $O B$ with the vertical line through $A$ . If $M$ is the point on the line $O B C$ such that $O M$ and $B C$ are of equal length, then the locus of point $M$ as $B$ varies is given by the equation

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Important Questions on Circle

MEDIUM

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

Let the normals at all the points on a given curve pass through a fixed point

(a, b)

. If the curve passes through

(3, - 3)

and

(4, - 2 \sqrt{2})

, given that

a - 2 \sqrt{2} b = 3

, then

(a^{2} + b^{2} + a b)

is equal to _____.

MEDIUM

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

y + c = 0

is a tangent to the circle

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

(a, 4)

, then

MEDIUM

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

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>Tangent and Normal of a Circle

Let

B

be the centre of the circle

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

Let the tangents at two points

P

and

Q

on the circle intersect at the point

A (3, 1) .

Then

8 (\frac{area Δ APQ}{area Δ BPQ})

is equal to .

EASY

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

If a line

y = m x + c

is a tangent to the circle

{(x - 3)}^{2} + y^{2} = 1

and it is perpendicular to a line

L_{1},

where

L_{1}

is the tangent to the circle

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

at the point

(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})

, then

HARD

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

Let the lengths of intercepts on

x

-axis and

y

-axis made by the circle

x^{2} + y^{2} + a x + 2 a y + c = 0,

(a < 0)

2 \sqrt{2}

and

2 \sqrt{5}

, respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line

x + 2 y = 0,

is equal to :

MEDIUM

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

If the area of the triangle formed by the

x

-axis, the normal and the tangent to the circle

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

at the point

(5, 7)

A,

then

24 A

is equal to _________.

MEDIUM

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

P (- 9, - 1)

is a point on the circle

x^{2} + y^{2} + 4 x + 8 y - 38 = 0

, then find equation of the tangent drawn at the other end of the diameter drawn through

P

MEDIUM

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

Let the tangents drawn to the circle,

x^{2} + y^{2} = 16

from the point

P (0, h)

meet the

x

-axis at points

A

and

B

. If the area of

Δ A P B

is minimum, then positive value of

h

is:

MEDIUM

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

Let the tangents drawn from the origin to the circle,

x^{2} + y^{2} - 8 x - 4 y + 16 = 0

touch it at the points

A

and

B

. Then

{(A B)}^{2}

is equal to

MEDIUM

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

The radius of a circle, having minimum area, which touches the curve

y = 4 - x^{2}

and the lines,

y = |x|

is:

HARD

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

If a tangent to the circle

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

intersects the coordinate axes at distinct points

P

and

Q,

then the locus of the mid-point of

P Q

is:

EASY

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

The area of a circle having the lines

3 x - 4 y + 4 = 0

and

6 x - 8 y - 7 = 0

as two of its tangents, is

HARD

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

Let the tangent to the circle

x^{2} + y^{2} = 25

at the point

R (3, 4)

meet

x

-axis and

y

-axis at point

P

and

Q

, respectively. If

r

is the radius of the circle passing through the origin

O

and having centre at the incentre of the triangle

O P Q,

then

r^{2}

is equal to

EASY

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

Find the equations of the tangents drawn to the circle

x^{2} + y^{2} = 50

at the points where the line

x + 7 = 0

meets it.

EASY

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

Consider the circle

C

that passes through the points

(1, 0)

and

(0, 1)

having the smallest area. Then, the equation of the tangent to the circle

C

(0, 1)

EASY

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

Which of the following lines is a normal to the circle

x^{2} + y^{2} - 2 x - 10 y + 6 = 0

HARD

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

Let

C

be the circle concentric with the circle,

2 x^{2} + 2 y^{2} - 6 x - 10 y = 183

and having area

{(\frac{1}{10})}^{th}

of the area of this circle. Then a tangent to

C,

parallel to the line,

3 x + y = 0

makes an intercept on the

y

-axis, which is equal to

EASY

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

The tangent and the normal lines at the point

(\sqrt{3}, 1)

to the circle

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

and the

x

-axis form a triangle. The area of this triangle (in square units) is:

EASY

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

Equation of the tangent to the circle, at the point

(1, - 1)

, whose center, is the point of intersection of the straight lines

x - y = 1

and

2 x + y = 3

is:

Important Questions on Circle

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