If the lengths of the tangents drawn from P to the circles x^2+y^2-2 x+4 y-20=0 and x^2+y^2-2 x-8 y+1=0 are in the ratio 2: 1 , then the locus P is

Step-by-step solution

We know that, length of tangent drawn from (x₁, y₁ ) to the circle x^2+y^2+2 g x+2 f y+c=0 is x₁^2+y₁^2+2 g x₁+2 f y₁+c Let P(h, k) According to the question, aligned & h^2+k^2-2 h+4 k-20 h^2+k^2-2 h-8 k+1 = 2 1 & h^2+k^2-2 h+4 k-20=4 (h^2+k^2-2 h-8 k+1 ) & h^2+k^2-2 h+4 k-20=4 h^2+4 k^2-8 h-32 k+4 & 3 h^2+3 k^2-6 h-36 k+24=0 & h^2+k^2-2 h-12 k+8=0 aligned Locus of point P is x^2+y^2-2 x-12 y+8=0 .