Question

The asymptotes of the hyperbola xy = hx + ky are –

• A. x = k, y = h
• B. x = h, y = k
• C. x = h, y = h
• D. x = k, y = k

Answer 1

Answer: (A)

x = k, y = h

Answer 2

The equations of asymptotes are

xy – hx – ky + l = 0 … (1)

where l is a constant and (1) represents a pair of straight line.

Here A = 0, B = 0, C = l, 2H = 1, 2G = –h and 2F = –k.

Then ABC + 2FGH – AF² – BG² – CH² = 0

Ž 0 + 2 $\left( - \frac{k}{2} \right)$ $\left( \frac{1}{2} \right)$ – 0 – 0 – l . $\frac{1}{4}$ = 0

Ž $\frac{hk}{4}$ = $\frac{\lambda}{4}$ Ž l = hk

Hence, on putting l = hk in equation (1), we get

xy – hx – ky + hk = 0

(x – k) (y – h) = 0

Asymptotes are x = k, and y = h.

Hence (1) is the correct answer.