If the circles x^2+y^2-4 x+2 f y+1=0 and x^2+y^2+2 gx -4 y -1=0 cut orthogonally, then r ₁^2+ r ₂^2-8=

Step-by-step solution

Let S₁: x^2+y^2-4 x+2 f y+1=0 ...(i) S₂: x^2+y^2+2 g x-4 y-1=0 ...(ii) aligned & g ₁=2, f ₁= f , c ₁=1 & ~g ₂= g , f ₂=2, c ₂=-1 aligned Circle (i) & (ii) are orthogonal : 2 (g₁ g₂+f₁ f₂ )=c₁+c₂ and r₁^2+r₂^2=d^2 where d is distance between the centers aligned & Now, 2 ( f ₁ f ₂+ g ₁ ~g ₂ )= c ₁+ c ₂ & 2(2 f +2 ~g )=1-1 f =- g & Also, r ₁ ^2+ r ₂ ^2= d ^2 aligned aligned & r₁^2+r₂^2=(g-2)^2+(g+2)^2 & r₁^2+r₂^2=2 g^2+8 r₁^2+r₂^2-8=2 g^2 aligned