If (a, b) and (c, d) are the internal and external centres of similitudes of the circles x^2+y^2+4 x-5=0 and x^2+y^2-6 y+8=0 respectively, then (a+d)(b+c)=

Step-by-step solution

Given equation of two circle x^2+y^2+4 x-5=0 and x^2+y^2-6 y+8=0 (x+2)^2+y^2=3^2 and, x^2+(y-3)^2=1 So, C ₁=(-2,0), r₁=3 ; C ₂=(0,3), r₂=1 aligned & Now, internal centre = ( r₂ x₁+r₁ x₂ r₁+r₂ , r₂ y₁+r₁ y₂ r₁+r₂ ) & = ( -2+0 4 , 0+9 4 ) (a, b)= ( -1 2 , 9 4 ) aligned aligned & and external centre = ( x₂ r₁-r₂ x₁ r₁-r₂ , r₁ y₂-r₂ y₁ r₁-r₂ ) & = ( 0+2 3-1 , 3 3-0 3-1 ) ( c , d )= (1, 9 2 ) aligned Now, (a+d)(b+c)= ( -1 2 + 9 2 ) ( 9 4 +1 )=4 13 4 =13