The lengths of the tangents from any point on the circle x 2 + y 2 + 8 x + 1 = 0 to the circles x 2 + y 2 + 7 x + 1 = 0 and x 2 + y 2 + 4 x + 1 = 0 are in the ratio

Step-by-step solution

Let the point on x 2 + y 2 + 8 x + 1 = 0 is h , k ⇒ h 2 + k 2 + 8 h + 1 = 0 Now ratio of the lengths of the tangents from h , k to the circles x 2 + y 2 + 7 x + 1 = 0 and x 2 + y 2 + 4 x + 1 = 0 is h 2 + k 2 + 7 h + 1 h 2 + k 2 + 4 h + 1 = 7 h - 8 h 4 h - 8 h = 1 2 = 1 : 2