If P x 1 , y 1 is a point such that the lengths of the tangents from it to the circles x 2 + y 2 - 4 x - 6 y - 12 = 0 and x 2 + y 2 + 6 x + 18 y + 26 = 0 are in the ratio 2 : 3 , then the locus of P is

Step-by-step solution

Given: C 1 ≡ x 2 + y 2 - 4 x - 6 y - 12 = 0 C 2 ≡ x 2 + y 2 + 6 x + 18 y + 26 = 0 Now, we know that length of tangent drawn from any point to circle is given by S 1 . Now, it is given that, S 1 S 2 = 2 3 ⇒ S 1 S 2 = 4 9 ⇒ 9 S 1 - 4 S 2 = 0 ⇒ 9 h 2 + k 2 - 4 h - 6 k - 12 - 4 h 2 + k 2 + 6 h + 18 k + 26 = 0 ⇒ 5 h 2 + 5 k 2 - 60 h - 126 k - 212 = 0 ⇒ h 2 + k 2 - 12 h - 126 5 k - 212 5 = 0 Therefore, locus is x 2 + y 2 - 12 x - 126 5 y - 212 5 = 0 .