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

Option: 1
$-7$

Option: 2
$-5$

Option: 3
$5$

Option: 4
$7$

Answers (1)

best_answer

$S(5,0)$ is the focus $\mathrm { \therefore a e=5 ......(i)}$

$\mathrm { 5 x=9\: \: i.e., x=\frac{9}{5} \: \: is \: \: the\: \: directrix \therefore \frac{a}{e}=\frac{9}{5}......(ii) }$

From (i) and (ii), we get $\mathrm { a=3 \: \: and\: \: e=\frac{5}{3} }$

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

Hence, $\mathrm {a^2-b^2=9-16=-7 }$

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Divya Prakash Singh

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