First slide

Hyperbola in conic sections

Question

The asymptotes of the hyperbola xy = hx + ky are

Moderate

A

x-k=0 and y-h=0
B

x+h=0 and y+k=0
C

x - k =0 and y + h = 0
D

x+k= 0 and y-h=0

Solution

The given hyperbola is
$x y - h x - l q = 0 - - - - - (1)$
The equation of asymptotes is given by
$x y - h x - k y + c = 0 - - - - (2)$
Equation (2) gives a pair of straight lines. So,
$|\begin{array}{ccc} A & H & G \\ H & B & F \\ G & F & C \end{array}| = 0$
$or |\begin{array}{ccc} 0 & 1 / 2 & - h / 2 \\ 1 / 2 & 0 & - k / 2 \\ - h / 2 & - k / 2 & c \end{array}| = 0$
$\begin{array}{l} or \frac{hk}{8} + \frac{hk}{8} - \frac{c}{4} = 0 \\ or c = hk \end{array}$
Hence, the asymptotes are
$\begin{array}{ll} x y - h x - k y + h k = 0 \\ or (x - k) (y - h) = 0 \end{array}$

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