उत्तर (51)

समाधान

फोकस $\equiv( \pm 5,0) ; \frac{2 b^{2}}{a^{2}}=\sqrt{50}$

$a=5 \quad b^{2}=\frac{5 \sqrt{2} a}{2}$

$b^{2}=a^{2}\left(1-e^{2}\right)=\frac{5 \sqrt{2} a^{2}}{2}$

$\Rightarrow a\left(1-e^{2}\right)=\frac{5 \sqrt{2}}{2}$

$\Rightarrow \frac{5}{e}\left(1-e^{2}\right)=\frac{5 \sqrt{2}}{2}$

$\Rightarrow \sqrt{2}-\sqrt{2} e^{2}\neq e$

$\Rightarrow \sqrt{2} e^{2}+e-\sqrt{2}\neq 0$

$\Rightarrow \sqrt{2} e^{2}+2 e-e-\sqrt{2}=0$

$\Rightarrow \sqrt{2} e(e+\sqrt{2})-1(1+\sqrt{2})=0$

$\Rightarrow(e+\sqrt{2})(\sqrt{2} e-1)=0$

$\therefore e \neq-\sqrt{2} ; e=\sqrt{\frac{1}{2}}$

$\frac{x^{2}}{b^{2}}-\frac{y^{2}}{a^{2} b^{2}}=1 \quad a=5 \sqrt{2}$

$b=5$

$a^{2} b^{2}=b^{2}\left(e_1^{2}-1\right) \Rightarrow e_1^{2}=51$