If P is a point such that the ratio of the square of the lengths of the tangents from P to the circles x^2+y^2+2 x-4 y-20=0 and x^2+y^2-4 x+2 y-2 y-44=0 is 2: 3 , then the locus of P is a circle with centre :

Step-by-step solution

Let co-ordinates of P be (x₁, y₁ ) . Given that, x^2+y^2+2 x-4 y-20=0 ...(i) and x^2+y^2-4 x+2 y-44=0 ...(ii) Length of the tangent from P to Eq. (i) =x₁^2+y₁^2+2 x₁-4 y₁-20 ...(iii) Length of the tangent from P to Eq. (ii) =x₁^2+y₁^2-4 x₁+2 y₁-44 ...(iv) Given that ratio of lengths of tangent = 2 3 x₁^2+y₁^2+2 x₁-4 y₁-20 x₁^2+y₁^2-4 x₁+2 y₁-44 = 2 3 3 x₁^2+3 y₁^2+6 x₁-12 y₁-60=2 x₁^2+2 y₁^2-8 x₁+4 y₁-88 x₁^2+y₁^2+14 x₁-16 y₁+28=0 Locus of points P is x^2+y^2+14 x-16 y+28=0 Centre of the circle is (-7,8) .