Correct option is: (4) 2587
\(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\)
Directrix : \(x=\frac{\sqrt{10}}{9}=\frac{a}{e}\) .....(i)
Focus: \((a e, 0) \equiv a e=\sqrt{10}\) .....(ii)
\((i) \times (ii)\)
\(a^{2}=\frac{10}{9}\)
Dividing equation (ii) by (i) we get,
\(e^{2}=9 \Rightarrow e=3\)
Also, \(e^{2}=9=1+\frac{9 b^{2}}{10}\)
\(\Rightarrow b^{2}=\frac{80}{9}\)
\(I=2 \times \frac{b^{2}}{a}=\frac{2 \times \frac{80}{9}}{\frac{\sqrt{10}}{3}}=\frac{160}{3 \sqrt{10}}\)
Now, \(9\left(e+l^{2}\right)=9\left[3+\frac{25600}{9 \times 10}\right]\)
\(9\left(3+\frac{160}{3 \sqrt{10}}\right)=27+2560\)
= 2587
