Let a and b respectively be the semi-transverse and semi-conjugate...

Let

a

and

b

respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation

9{e}^{2} - 18e + 5 = 0

. If $

S\left(5, 0\right)

is a focus and

5x = 9

is the corresponding directrix of this hyperbola, then ${a}^{2} - {b}^{2} is equal to

Options

A $- 7$
B $- 5$
C $5$
D $7$

Correct answer

A. $- 7$

Step-by-step solution

Given that focus of hyperbola is $S\left(5, 0\right)$ ⇒ ae = 5 $......\left(i\right) Therefore, Directrix$ x = $\dfrac{a}{e}$ ⇒ $\dfrac{a}{e} = \dfrac{9}{5} ......\left(ii\right) From \left(i\right)$ and \left(ii\right), we get $⇒ {a}^{2} = 9$

⇒ a = 3 and $e =$ \dfrac{5}{3}

(It satisfies $9{e}^{2} - 18e + 5 = 0$ )

We also know that for an ellipse,

${b}^{2} = {a}^{2}\left({e}^{2} - 1\right)$

$⇒ {b}^{2} = 16$

$⇒ {a}^{2} - {b}^{2} = 9 - 16 = - 7$

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