The locus of the point of intersection of the tangents at the extremities of a chord of the circle x 2 + y 2 = r 2 which touches the circle x 2 + y 2 + 2 r x = 0 is

Step-by-step solution

Let, point of intersection of the tangents is h , k . Chord of contact through h , k for the circle x 2 + y 2 = r 2 is h x + k y - r 2 = 0 which touches the circle x 2 + y 2 + 2 r x = 0 ⇒ The distance of h x + k y - r 2 = 0 from the centre - r , 0 of circle x 2 + y 2 + 2 r x = 0 is equal to the radius r of the circle x 2 + y 2 + 2 r x = 0 ⇒ h - r + k 0 - r 2 h 2 + k 2 = r ⇒ h + r = h 2 + k 2 ⇒ x 2 + 2 r x + r 2 = x 2 + y 2 ⇒ y 2 = 2 r x + r 2 is the required locus