If the circles ((x+a)^2+(y+b)^2=a^2 ) and ((x+c)^2+(y+d)^2=d^2 ) cut orthogonally, then (b(b-2 d)= )

Step-by-step solution

Given equation of circles are ( aligned (x+a)^2+(y+b)^2 & =a^2 x^2+y^2+2 a x+2 b y+b^2 & =0 (i) (x+c)^2+(y+d)^2 & =d^2 x^2+y^2+2 c x+2 d y+c^2 & =0 (ii) aligned ) From Eq. (i) (g₁=a ; f₁=b, c₁=b^2 ) From Eq. (ii) (g₂=c ; f₂=d ; c₂=c^2 ) If circles (i) and (ii) are orthogonal, ( array rlrl & & 2 [g₁ g₂+f₁ f₂ ] & =c₁+c₂ & 2 & 2(a c+b d) & =b^2+c^2 & & 2 a c+2 b d & =b^2+c^2 & & 2 a c-c^2 & =b^2-2 b d & & c(2 a-c) & =b(b-2 d) & & b(b-2 d) & =c(2 a-c) array ) ( ) Hence, answer is (b).