Question

Find the asymptotes of the following hyperbolas and also the equations to their conjugate hyperbolas.
$y^2-xy-2x^2-5y+x-6=0.$

Answer 1

Given equation of the hyperbola: $y^2-xy-2x^2-5y+x-6=0$

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: $y^2-xy-2x^2-5y+x+K=\mathrm{0...}\left(i\right)$

Factorising the 2nd degree terms, gives

$y^2-xy-2x^2=y^2-2xy+xy-2x^2=(y-2x)(y+x)$

Therefore the asymptotes have equations $y-2x+l=0$ and $y+x+m=0$ and their combined equation will be $(y-2x+l)(y+x+m)=\mathrm{0...}\left(ii\right)$.

Equations (i) & (ii) represent the same pair of lines.

Comparing the coefficient of x, y and constant term gives,

$m+l=-5,-2m+l=1$ & $lm=K$,

which gives $l=-3$, and $m=-2$

Therefore the asymptotes are $y-2x-3=0$ and $y+x-2=0$ and their combined equation is $y^2-xy-2x^2-5y+x+6=0$.

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 = $2(y^2-xy-2x^2-5y+x+6)-(y^2-xy-2x^2-5y+x-6)$

$\Rightarrow$ equation of conjugate hyperbola = $y^2-xy-2x^2-5y+x+18=0$