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Let $A B C D$ be a quadrilateral in which $A B$ is parallel to $C D, A B$ is perpendicular to $A D$ and $A B = 3 C D .$ The area of the quadrilateral $A B C D$ is $4$ sq. units. The diameter (in units) of a circle touching all the sides of the quadrilateral is

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Important Questions on Circle

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If a variable line

3 x + 4 y - λ = 0

is such that the two circles

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

and

x^{2} + y^{2} - 18 x - 2 y + 78 = 0

are on its opposite sides, then the set of all values of

λ

is the interval :

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

The numbers of integral values of

k

for which the line,

3 x + 4 y = k

intersects the circle,

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

at two distinct points is....

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If the length of the chord of the circle,

x^{2} + y^{2} = r^{2} (r > 0)

along the line,

y - 2 x = 3

r

, then

r^{2}

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If the circle

x^{2} + y^{2} - 6 x + 2 y = 28

cuts off a chord of length

λ

units on the line

2 x - 5 y + 18 = 0,

then the value of

λ

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If the tangent at the point

P

on the circle

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

meets the straight line.

5 x - 2 y + 6 = 0

at a point

Q

on the

y

-axis, then the length of

P Q

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If a line drawn from a fixed point

M (a, b)

cuts the circle

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

C

and

D,

then

M C \times M D

is equal to

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

The diameter of the circle, whose Centre lies on the line

x + y = 2

in the first quadrant and which touches both the lines

x = 3

and

y = 2

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

The line

2 x - y + 1 = 0

is a tangent to the circle at the point

(2, 5)

and the centre of the circle lies on

x - 2 y = 4 .

Then, the radius of the circle is:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

For

a = \sqrt{2}

, if a tangent is drawn to a suitable conic (Column 1) at the point of contact

(- 1, 1)

, then which of the following options is the only Correct combination for obtaining its equation?

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

The sum of the squares of the lengths of the chords intercepted on the circle,

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

by the lines,

x + y = n, n \in N,

where

N

is the set of all natural numbers is:

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If the line

x + y + 1 = 0

intersects the circle

x^{2} + y^{2} + x + 3 y = 0

at two points

A

and

B,

then the centre of the circle which passes through the points

A, B

and the point of intersection of the tangents drawn at

A

and

B

to the given circle is

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If two parallel chords of a circle, having diameter

4

units, lie on the opposite sides of the center and subtend angles

\cos^{- 1} (\frac{1}{7})

and

\sec^{- 1} (7)

at the center respectively, then the distance between these chords is:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

The length of the chord joining the points in which the straight line

\frac{x}{3} + \frac{y}{4} = 1

cuts the circle

x^{2} + y^{2} = \frac{169}{25}

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

The area of the triangle formed by joining the origin to the point of intersection of the line

x \sqrt{5} + 2 y = 3 \sqrt{5}

and circle

x^{2} + y^{2} = 10

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

Let

A B C D

be a square of side of unit length. Let a circle

C_{1}

centered at

A

with unit radius is drawn. Another circle

C_{2}

which touches

C_{1}

and the lines

A D

and

A B

are tangent to it, is also drawn. Let a tangent line from the point

C

to the circle

C_{2}

meet the side

A B

E

. If the length of

E B

α + \sqrt{3} β

, where

α, β

are integers, then

α + β

is equal to ________.

EASY

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

If the length of the common chord of two circles

x^{2} + y^{2} + 8 x + 1 = 0

and

x^{2} + y^{2} + 2 μ y - 1 = 0

2 \sqrt{6},

then the values of

μ

are:

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

Consider the circle with centre at the point

(1, 2)

and having the line

x = y

as a tangent. The area of the circle is

HARD

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

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>Interaction between Circle and a Line

Let

C

be the circle

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

in the

x y

- plane. For each

t \geq 0, l e t L_{t}

be the line passing through

(0, 1) and (t, 0)

. Note that

L_{t}

intersects

C

in two points, one of which is

(0, 1) .

Let

Q_{t}

be the other point. As

t

varies between

1 & 1 + \sqrt{2},

the collection of points

Q_{t}

sweeps out an area on

C

. The angle subtended by this arc at

(0, 0)

MEDIUM

Mathematics>Coordinate Geometry>Circle>Interaction between Circle and a Line

y + 3 x = 0

is the equation of a chord of the circle

x^{2} + y^{2} - 30 x = 0

, then the equation of the circle with this chord as diameter is :

Let ABCD be a quadrilateral in which AB is parallel to CD, AB is perpendicular to AD and AB=3CD. The area of the quadrilateral ABCD is 4 sq. units. The diameter (in units) of a circle touching all the sides of the quadrilateral is

Important Questions on Circle

Learn and Prepare for any exam you want !

Let $A B C D$ be a quadrilateral in which $A B$ is parallel to $C D, A B$ is perpendicular to $A D$ and $A B = 3 C D .$ The area of the quadrilateral $A B C D$ is $4$ sq. units. The diameter (in units) of a circle touching all the sides of the quadrilateral is