The locus of the mid-point of the chords of the hyperbola x 2 - y 2 = 4 , that touches the parabola y 2 = 8 x is

Step-by-step solution

Let midpoint of the chord be P h , k . Then, the equation of the chord will be x h - y k = h 2 - k 2 ⇒ y = h k x + k 2 - h 2 k … 1 Now, the line y = m x + c is tangent to y 2 = 4 a x if c = a m Hence, equation 1 is tangent to y 2 = 8 x if k 2 - h 2 k = 2 k h ⇒ y 2 x - x 3 = 2 y 2 ⇒ y 2 x - 2 = x 3