Let the parabola be y 2 = 4ax and the rectangular hyperbola be x 2 – y 2 = a 2 . Let P(h, k) be the point whose chord of contact with respect to the parabola y 2 = 4ax touches the rectangular hyperbola. The chord of contact of tangents from P(h, k) to the parabola y 2 = 4ax is

ky = 2a (x + h)

⇒ 2ax – ky + 2ah = 0 … (1)

This touches the rectangular hyperbola x 2 – y 2 = a 2 . So, it should be of the form

x sec θ – y tan θ = a … (2)

From equations (1) and (2), we get

= =

⇒ sec θ = – and tan θ = –

⇒ sec 2 θ – tan 2 θ = –

⇒ 1 = –

⇒ 4h 2 + k 2 = 4a 2 ∴ The locus of (h, k) is 4x

2 + y 2 = 4a 2 which represents an ellipse with centre at (0, 0) and axes same as that of the hyperbola x 2 – y 2 = a 2 .

Hence is correct answer.