Question

If the axis of a parabola and a rectangular hyperbola is same and the vertex of the parabola is same as the centre of rectangular hyperbola, then the locus of the point whose chord of contact with respect to the parabola touches the rectangular hyperbola is a/an-

• A. Ellipse
• B. Hyperbola
• C. Circle
• D. Parabola

Answer 1

Answer: (A)

Ellipse

Answer 2

Let the parabola be y² = 4ax and the rectangular hyperbola be x² – y² = a². Let P(h, k) be the point whose chord of contact with respect to the parabola y² = 4ax touches the rectangular hyperbola. The chord of contact of tangents from P(h, k) to the parabola y² = 4ax is

ky = 2a (x + h)

Ž 2ax – ky + 2ah = 0 … (1)

This touches the rectangular hyperbola x² – y² = a². So, it should be of the form

x sec q – y tan q = a … (2)

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

$\frac{2a}{\sec\theta}$ = $\frac{- k}{- \tan\theta}$ = $\frac{2ah}{- a}$

Ž sec q = – $\frac{a}{h}$ and tan q = – $\frac{k}{2h}$

Ž sec² q – tan² q = $\frac{a^{2}}{h^{2}}$ – $\frac{k^{2}}{4h^{2}}$

Ž 1 = $\frac{a^{2}}{h^{2}}$–$\frac{k^{2}}{4h^{2}}$

Ž 4h² + k² = 4a²

\ The locus of (h, k) is 4x² + y² = 4a² which represents an ellipse with centre at (0, 0) and axes same as that of the hyperbola x² – y² = a².

Hence (1) is correct answer.