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The locus of the midpoints of the chords of the circle $x^{2} + y^{2} - a x - b y = 0$ which subtend a right angle at $(\frac{a}{2}, \frac{b}{2})$ is

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Important Questions on Circle

MEDIUM

KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 18

A tangent at a point on the circle

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

intersects a concentric circle

C

at two points

P

and

Q

. The tangents to the circle

X

P

and

Q

meet at a point on the circle

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

then the equation of circle is

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KVPY Aptitude Test - Stream SA

IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 19

The chord of contact of tangents from a point

P

to a circle passes through

Q

. If

l_{1}

and

l_{2}

are the lengths of the tangents from

P

and

Q

to the circle, then

P Q

is equal to

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 20

If the circle

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

is touched by

y = x

P

such that

O P = 6 \sqrt{2},

then the value of

c

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 21

The circle

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

cuts the line joining the points

A (1, 0)

and

B (3, 4)

in two points

P

and

Q .

Let

\frac{B P}{P A} = α

and

\frac{B Q}{Q A} = - β .

Then

α

and

β

are roots of the quadratic equation

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 22

The equation of circumcircle of an equilateral triangle is

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

and one vertex of the triangle is

(1, 1),

the equation of incircle of the triangle is

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 23

Let the base

A B

of a triangle

A B C

be fixed and the vertex

C

lie on a fixed circle of radius

r

. Lines through

A

and

B

are drawn to intersect

C B

and

C A,

respectively, at

E

and

F

such that

C E : E B = 1 : 2

and

C F : F A = 1 : 2

If the point of intersection

P

of these lines lies on the median through

A B

for all positions of

A B

then the locus of

P

HARD

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IMPORTANT

Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 24

Six points

(x_{i}, y_{i}), i = 1, 2 \dots \dots \dots, 6

are taken on the circle

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

such that

\sum_{i = 1}^{6} x_{i} = 8

and

\sum_{i = 1}^{6} y_{i} = 4

, the line segment joining orthocentre of a triangle made by any three points and the centroid of the triangle made by other three points passes through a fixed point

(h, k),

then

h + k

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Practice Book for KVPY Aptitude Test - Stream SA Mathematics>Chapter 11 - Circle>Exercise>Q 25

A circle with radius

|a|

and centre on

y

-axis slides along it and a variable lines through

(a, 0)

cuts the circle at points

P

and

Q .

The region in which the point of intersection of tangents to the circle at points

P

and

Q

lies is represented by

The locus of the midpoints of the chords of the circle x2+y2-ax-by=0 which subtend a right angle at a2, b2 is

Important Questions on Circle

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The locus of the midpoints of the chords of the circle $x^{2} + y^{2} - a x - b y = 0$ which subtend a right angle at $(\frac{a}{2}, \frac{b}{2})$ is