Question

If foci of hyperbola lie on $y=x$ and one of the asymptote is $y=2x$, then equation of the hyperbola, given that is passes through $(3,4)$ is :

• A. $x^2-y^2-\frac{5}{2}xy+5=0$
• B. $2x^2-2y^2+5xy+5=0$
• C. $2x^2+2y^2-5xy+10=0$
• D. None of these

Answer 1

The correct option is C $2x^2+2y^2-5xy+10=0$

Foci of hyperbola lie on $y=x$.
So, the equation of transverse axis is $y-x=0$.
Transverse axis of hyperbola bisects the asymptote
$\Rightarrow$ equation of other asymptote is $y=\frac{x}{2}$
or,$x=2y$
$\Rightarrow$ Equation of hyperbola is $(y-2x)(x-2y)+k=0$
Since, it passes through $(3,4)$
$\Rightarrow k=-10$
Hence, required equation is
$2x^2+2y^2-5xy+10=0$