The equation of the circle whose diameter is common chord to the circles

Options

A. ` x ^ 2 + y ^ 2 - ( 2ab^ 2 ) /( a ^ 2 + b ^ 2) x + ( 2a ^ 2 b) /( a ^ 2 + b ^ 2 ) y + c = 0 `
B. ` x ^ 2 + y^ 2 - ( 2ab ^ 2 ) /( a ^ 2 + b^ 2 )x + ( 2a^ 2 b ) /( a ^ 2 + b ^ 2 ) y + c = 0 `
C. ` x^ 2 + y ^ 2 + ( 2ab ^ 2 ) /( a ^ 2 + b ^ 2 ) x + ( 2a^ 2 b ) /( a ^ 2 + b^ 2 ) y + c = 0 `
D. ` x ^ 2 + y^ 2 + ( 2ab ^ 2 ) /( a ^2 + b ^ 2 ) x - ( 2a^ 2 b) /( a ^ 2 + b^ 2 ) y +c = 0`

Answer

C

Explanation

Let ` S _ 1 -= x ^ 2 + y^ 2 + 2ax + c = 0 `
and ` S _ 2 -= x ^ 2 = y ^ 2 + 2by + c = 0 `
Equation of common chord as a diameter of the third circle
` S _ 1 - S _ 2 = 0 `
` a x - by = 0 `
` y = ( ax ) / ( b ) " " ` ... (i)
On putting the value of y in Eq. (i) , we get
` x ^ 2 + ( ( a^ 2 x ^ 2 )/ ( b ^ 2 )) + 2ax + c = 0 `
` ( a ^ 2 + b ^ 2 ) x ^ 2 + 2 ab ^ 2 x + cb ^ 2 = 0 `
Let ` x _ 1 and x _ 2 ` be the roots of the equation.
` therefore x _ 1 + x _ 2 = ( - 2ab^ 2 ) /( a ^ 2 + b ^ 2 ) and x _ 1 x _ 2 = ( cb ^ 2 ) /( a ^ 2 + b^ 2 ) `
Similarly, ` ( a ^ 2 + b ^ 2 ) y ^2 + 2 ba ^ 2 y + ca ^ 2 = 0 `
Let ` y _ 1 and y _ 2 ` be the roots of the equation
` therefore y _1 + y _ 2 = ( - 2a^ 2 b ) /( a ^ 2 + b ^ 2 ) `
and ` y _ 1 y _ 2 = ( ca ^ 2 ) /( a ^ 2 + b ^ 2 ) `
Now, the required equation of third circle
` ( x - x _ 1 ) ( x - x _ 2 ) + ( y - y _ 1 ) ( y - y _ 2 ) = 0 `
` x ^ 2 - ( x _ 1 + x _ 2 ) x + x _ 1 x _ 2 + y ^ 2 - ( y _ 1 + y _ 2 ) y + y _1 y _ 2 = 0`
` x^ 2 + y^ 2 + ( 2ab^ 2 ) /( a ^ 2 + b ^ 2 ) * x + ( 2ba^ 2 ) /( a^ 2 + b^ 2 ) * y + ( cb^ 2 )/( a ^ 2 + b^ 2 ) + ( ca ^ 2 ) /( a ^ 2 + b ^ 2 ) = 0 `
` therefore x ^ 2 + y ^ 2 + ( 2 ab ^ 2 ) /( a ^ 2 + b ^ 2 ) * x + ( 2a ^ 2 b ) /( a ^ 2 + b ^ 2 ) * y + ( c ( a^ 2 + b ^ 2 )) /( ( a ^ 2 + b ^ 2 ) ) = 0 `
` rArr x ^ 2 + y^ 2 + ( 2 ab ^ 2 ) /( ( a ^ 2 + b ^ 2 )) * x + ( 2a^ 2 b ) /( ( a ^ 2 + b^ 2)) * y + c = 0`