Question

Find the equation to the hyperbola, whose eccentricity is $\frac{5}{4}$, focus is $(a,0)$ and whose directrix is $4x-3y=a$.

• A. $7y^2+24xy-12ax-3ay+15a^2=0$
• B. $7y^2+24xy+12ax+3ay+15a^2=0$
• C. $7y^2+24xy+24ax+6ay+15a^2=0$
• D. $7y^2+24xy-24ax-6ay+15a^2=0$

Answer 1

The correct option is D $7y^2+24xy-24ax-6ay+15a^2=0$

Using Hyperbola definition, $P{S}^2=e^2.P{M}^2$
$(x-a{)}^2+(y-0{)}^2=\frac{25}{16}{\left|\frac{4x-3y-a}{\sqrt{3^2+4^2}}\right|}^2$
$\Rightarrow 16(x^2+y^2-2ax+a^2)=16x^2+9y^2+a^2-24xy+6ya-8ax$
$\Rightarrow 7y^2+24xy-24ax-6ay+15a^2=0$