Question

If $5x+9=0$ is the directrix of the hyperbola $16x^2-9y^2=144,$ then its corresponding focus is

• A. $(-5,0)$
• B. $(5,0)$
• C. $(-\frac{5}{3},0)$
• D. $(\frac{5}{3},0)$

Answer 1

The correct option is A $(-5,0)$

Given,
Equation of hyperbola $\Rightarrow 16x^2-9y^2=144$
$\Rightarrow \frac{x^2}{9}-\frac{y^2}{16}=1$

Here $a=3,b=4$
Eccentricity is
$\Rightarrow e^2=1+\frac{16}{9}$
$\Rightarrow e=\frac{5}{3}$
Foci are $(\pm ae,0)\equiv (\pm 5,0)$
Now given directrix is $x=-\frac{9}{5}$
which is of form $x=-\frac{a}{e}$
Hence corresponding focus will be $(-ae,0)\equiv (-5,0)$