If the points of intersection of the ellipses x²+2 y²-6 x-12 y+23=0 and 4 x²+2 y²-20 x-12 y+35=0 lie on a circle of radius r and centre (a, b) , then the value of a b+18 r² is

Step-by-step solution

The circle through intersection of the ellipses is S₁ + S₂ = 0 . (1+4 )x^2 + (2+2 )y^2 - (6+20 )x - (12+12 )y + (23+35 ) = 0 . For a circle: 1+4 = 2+2 = 1 2 . Circle: 3x^2 + 3y^2 - 16x - 18y + 81 2 = 0 x^2 + y^2 - 16 3 x - 6y + 27 2 = 0 . Centre (a,b) = ( 8 3 , 3 ) , r^2 = 64 9 + 9 - 27 2 = 47 18 . ab + 18r^2 = 8 3 3 + 18 47 18 = 8 + 47 = 55 .