Question

A hyperbola has center $C$ and one focus at $P\left(6 , \, 8\right)$ . If its two directrices are $3x+4y+10=0$ and $3x+4y-10=0$ , then $CP=$

Answer 1

Distance between the directrices $=\frac{2 a}{e}=\left|\frac{10 + 10}{\sqrt{3^{2} + 4^{2}}}\right|=\frac{20}{5}=4$
$\Rightarrow a=2e$ .
If we calculate distance of $P$ from $3x+4y+10=0$ , it is
$\left|\frac{3 \times 6 + 4 \times 8 + 10}{\sqrt{3^{2} + 4^{2}}}\right|=12$
And from other directrix is
$\left|\frac{3 \times 6 + 4 \times 8 - 10}{\sqrt{3^{2} + 4^{2}}}\right|=8$
Therefore, $P$ is nearest to $3x+4y-10=0$
Distance of $P$ from this directrix is
$\Rightarrow ae-\frac{a}{e}=8$
$\Rightarrow 2e^{2}=10$
$\Rightarrow e=\sqrt{5}, \, a=2\sqrt{5}$
$\Rightarrow CP=ae=10$