We have ` 2x ^(2) + 2y ^(2) - 8x - 12 y - 9 =0 `
`rArr x ^(2) + y ^(2) - 4x - 6y - (9)/(2) = 0 rArr x ^(2) + y ^(2) + 2gx + 2 f y + c = 0 `
` " "` where ` g = - 2 , f =-3 and c = (-9)/(2)`.
Centre of this circle ` = (-g, - f )= (2,3 )`
` therefore ` the centre of the required circle = `C(2, 3 )`
Again , ` x ^(2) + y ^(2) + 2 gx + 2 fy + c = 0, ` where ` g = 4, f = 5 and c = - 7`
Centre of this circle = `(-g, - f )= (- 4, - 5)`.
So, the required circle passes through the point ` P(-4, - 5)`.
Radius of the required circle ` = CP = sqrt (( 2+ 4 ) ^(2) + ( 3+ 5 ) ^(2)) `
`" "= sqrt ( 36 + 64 ) = sqrt (100) = 10 `
Hence, the required equation of the circle is
` (x - 2 ) ^(2) + (y - 3 ) ^(2) = (10)^(2) rArr x ^(2) + y ^(2) - 4x - 6y - 87 = 0 `
