Prove that the length of the common chord of the two circles whose equations are x - a 2 + y - b 2 = c 2 and x - b 2 + y - a 2 = c 2 is 4 c 2 - 2 a - b 2 . Hence, find the condition that the two circles may touch.

Equations of given circles are S 1 = x - a 2 + y - b 2 = c 2 x 2 + y 2 - 2 a x - 2 b y + a 2 + b 2 - c 2 = 0 … . 1 S 2 = x - b 2 + y - a 2 = c 2 x 2 + y 2 - 2 b x - 2 a y + a 2 + b 2 - c 2 = 0 … 2 Equation of common chord of two circles S 1 = 0 a n d S 2 = 0 i s S 1 - S 2 = 0 Equation of the common chord 1 a n d 2 i s , 1 - 2 = 0 x 2 + y 2 - 2 a x - 2 b y + a 2 + b 2 - c 2 - x 2 + y 2 - 2 b x - 2 a y + a 2 + b 2 - c 2 = 0 x - y = 0 . . . 3 Let P Q , be a chord of a circle whose center is O . If OA , is perpendicular from the center on the chord then, From ∆ O A P , O P 2 = O A 2 + A P 2 P Q = 2 A P = 2 O P 2 - O A 2 Length of chord P Q = 2 radius 2 - lengh of perpendicular from centre to chord 2 Center of S 1 is a , b , and radius is c The perpendicular distance from x 1 , y 1 , to the line a x + b y + c = 0 is a x 1 + b y 1 + c a 2 + b 2 ∴ O A = a - b 2 P Q = 4 c 2 - 2 a - b 2 If two circles touch each other, then the length of the common chord will be zero. 2 c 2 = a - b 2 . This is the required condition for two circles to touch.

HARD

JEE Advanced

IMPORTANT

Earn 100

$A, B, C$ and $D$ are four points in a straight line prove that the locus of a point $P$ such that the angles $A P B$ and $C P D$ are equal, is a circle.

Important Questions on The Circle

EASY

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 28

A tangent is drawn to the circle

{(x - a)}^{2} + y^{2} = b^{2}

and a perpendicular tangent to the circle

{(x + a)}^{2} + y^{2} = c^{2}

find the locus of their point of intersection, and prove that the bisector of the angle between them always touches one or other of two fixed circle.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 29

In any circle prove that the perpendicular from any point of it on the line joining the points of contact of tangents is a mean proportional between the perpendiculars from the point upon the two tangents.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 30

From any point on the circle

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

tangents are drawn to the circle

x^{2} + y^{2} + 2 g x + 2 f y + c \sin^{2} α + (g^{2} + f^{2}) \cos^{2} α = 0;

Prove that the angle between them is

2 α

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 31

The angular points of a triangle are the points

(acos α, asin α), (acos β, a \sin β)

and

(a \cos γ, a \sin γ)

. Prove that the coordinates of the orthocentre of the triangle are

a (\cos α + \cos β + \cos γ)

and

a (\sin α + \sin β + \sin γ) .

Hence, prove that if

A, B, C

and

D

, be four points on a circle the orthocentres of the four triangles

A B C, B C D, C D A

and

D A B

, lie on a circle.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 32

A variable circle passes through the point of intersection

O

, of any two straight lines and cuts off from them the portions

O P

and

O Q

, such that

m . O P + n . O Q

, is equal to unity; prove that this circle always passes through a fixed point.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 33

Find the length of the common chord of the circles, whose equations are

{(x - a)}^{2} + y^{2} = a^{2}

and

x^{2} + {(y - b)}^{2} = b^{2}

, Prove that the equation to the circle whose diameter is this common chord is

(a^{2} + b^{2}) (x^{2} + y^{2}) = 2 a b (b x + a y) .

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 34

Prove that the length of the common chord of the two circles whose equations are

{(x - a)}^{2} + {(y - b)}^{2} = c^{2}

and

{(x - b)}^{2} + {(y - a)}^{2} = c^{2}

\sqrt{\{4 c^{2} - 2 {(a - b)}^{2}\} .}

Hence, find the condition that the two circles may touch.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 6 - The Circle>EXAMPLES XXII>Q 35

Find the length of the common chord of the circles $x^{2} + y^{2} - 2 a x - 4 a y - 4 a^{2} = 0$ and $x^{2} + y^{2} - 3 a x + 4 a y = 0$ . Find also the equations of the common tangents and show that the length of each is $4 a$ .

A,B,C and D are four points in a straight line prove that the locus of a point P such that the angles APB and CPD are equal, is a circle.

Important Questions on The Circle

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$A, B, C$ and $D$ are four points in a straight line prove that the locus of a point $P$ such that the angles $A P B$ and $C P D$ are equal, is a circle.