Find the asymptotes of the following hyperbolas and also the equations to their conjugate hyperbolas.
$8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y = 2 .$

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Solution

Given equation of the hyperbola: $8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y = 2$
The equation of a hyperbola and the combined equation of the asymptotes differ only in the constant term.
Therefore the combined equation of the asymptotes is of the form: $8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y + K = 0... (i)$
Factorising the 2nd degree terms, gives
$8 x^{2} + 10 x y - 3 y^{2} = 8 x^{2} + 12 x y - 2 x y - 3 y^{2} = (4 x - y) (2 x + 3 y)$
Therefore the asymptotes have equations $4 x - y + l = 0$ and $2 x + 3 y + m = 0$ and their combined equation will be $(4 x - y + l) (2 x + 3 y + m) = 0... (i i)$ .
Equations (i) & (ii) represent the same pair of lines.
Comparing the coefficient of x, y and constant term gives,
$4 m + 2 l = - 2, - m + 3 l = 4, l m = K$
which gives $l = 1$ , and $m = - 1$
Therefore the asymptotes are $4 x - y + 1 = 0$ and $2 x + 3 y - 1 = 0$ and their combined equation is $8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y - 1 = 0$ .
Now using the fact that: equation of hyperbola + equation of conjugate hyperbola = 2x(equation of asymptotes)
$\Rightarrow$ equation of conjugate hyperbola = 2x(equation of asymptotes) - equation of hyperbola
$\Rightarrow$ equation of conjugate hyperbola = $2 (8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y - 1) - (8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y - 2)$
$\Rightarrow$ equation of conjugate hyperbola = $8 x^{2} + 10 x y - 3 y^{2} - 2 x + 4 y = 0$

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