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10-2. Parabola, Ellipse, Hyperbola
normal
Let $H : \frac{ x ^{2}}{ a ^{2}}-\frac{y^{2}}{ b ^{2}}=1$, a $>0, b >0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2 \sqrt{2}+\sqrt{14})$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of $a^{2}+b^{2}$ is equal to
A
$89$
B
$90$
C
$87$
D
$88$
(JEE MAIN-2022)
Solution
$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$
Given $e^{2}=1+\frac{b^{2}}{a^{2}} \Rightarrow \frac{11}{4}=1+\frac{b^{2}}{a^{2}} \Rightarrow b^{2}=\frac{7}{4} a^{2}$
$\therefore \frac{x^{2}}{(a)^{2}}-\frac{y^{2}}{\left(\frac{\sqrt{7}}{2} a\right)^{2}}=1$ Now given
$2 a+2 \cdot \frac{\sqrt{7} a}{2}=4(2 \sqrt{2}+\sqrt{14})$
$a(2+\sqrt{7})=4 \sqrt{2}(2+\sqrt{7})$
$a=4 \sqrt{2} \Rightarrow a^{2}=32$
$b^{2}=\frac{7}{4} \times 16 \times 2=56$
Standard 11
Mathematics
