If the circles x²+y²-2 x-2 y-7=0 and x²+y²+4 x+2 y+k=0 cut orthogonally, then the length of the common chord of the circles is

Step-by-step solution

Given circles are aligned &S₁ x²+y²-2 x-2 y-7=0 &S₂ x²+y²+4 x+2 y+k=0 aligned Here, g₁=-1, f₁=-1, c₁=-7 g₂=2, f₂=1, c₂=k Equation of common chord is S₁-S₂=0 gathered x²+y²-2 x-2 y-7-x²-y²-4 x -2 y-k=0 -6 x-4 y-7-k=0 gathered Since, S₁ and S₂ cut orthogonally. aligned & & 2 (g₁ g₂+f₁ f₂ ) &=c₁+c₂ & 2(-2-1) &=-7+k & &-6+7 &=k & & k &=1 & & & & & & aligned Now, length of the common chord r₁= 1+1+7 =3 c₁=(1,1) Let C₁ M= perpendicular distance from centre C₁(1,1) to the common chord aligned C₁ M &= |6+4+8| 6²+4² = |18|