Let a and b respectively be the semi-transverse and semi-conjugate axes of a hyperbola who…

Question

Let $a$ and $b$ respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation $9e^2-18e+5=0$. If $S(5,0)$ is a focus and $5x=9$ is the corresponding directrix of this hyperbola, then $a^2-b^2$ is equal to:

Accepted Answer

$9e^2-18e+5=0\Rightarrow(3e-1)(3e-5)=0\Rightarrow e=\dfrac53$ (taking $e>1$ for a hyperbola).

$ae=5$ and $\dfrac ae=\dfrac95\Rightarrow a^2=9\Rightarrow a=3$; $e=\dfrac53$.

$$b^2=a^2(e^2-1)=9\left(\frac{25}9-1\right)=16.$$

$a^2-b^2=9-16=-7.$

Solutions · 1