If S 2 x^2+2 y^2-8 x+8 y-7=0 is the circle passing through the points of intersection of the circles x^2+y^2+ kx - ky +1=0 and x ^2+ y ^2- kx + ky -2=0 , then the length of the tangent drawn from the point (k, k) to the circle S is

Step-by-step solution

The equation of circle through points of intersection of S₁, S₂ is: aligned & S ₁+ ( S ₂- S ₁ )=0 & x^2+y^2+k x-k y+1+ (x^2+y^2-k x+k y-2-x^2 . & . -y^2-k x+k y-1 )=0 & x^2+y^2+k x-k y+1+ (-2 k x+2 k y-3)=0 & x^2+y^2+(1-2 ) k x-(1-2 ) k y+(1-3 )=0 aligned ...(i) Given equation of circle through points of intersection of S ₁ & ~S ₂ is 2 x^2+2 y^2-8 x+8 y-7=0 or, x^2+y^2-4 x+4 y- 7 2 =0 ...(ii) Comparing eq ^ n (i) & (ii), we get k(1-2 )=-4 &-k(1-2 )=4 & 1-3 =- 7 2 3 = 9 2 = 3 2 In PMC , PM ^2+ CM ^2= CP ^2 aligned &