From the point A 0 , 3 on the circle x 2 + 4 x + y - 3 2 = 0 , a chord A B is drawn and extended from point B to a point M such that A M = 2 A B . The perimeter of the locus of M is p π units. Then, the value of p is

Step-by-step solution

The equation of the given circle is x 2 + y 2 + 4 x - 6 y + 9 = 0 … … . i Also, given that, A M = 2 A B or A B = B M Let, the coordinates of M be h , k . Then, B is the midpoint of A M . Therefore, B ≡ 0 + h 2 , 3 + k 2 ≡ h 2 , k + 3 2 As B lies on the circle, we have, from i , h 2 + k 2 + 8 h - 6 k + 9 = 0 ⇒ locus is a circle with radius 4 ⇒ the perimeter of the locus is 8 π units ⇒ p = 8