Let the eccentricity of the hyperbola H : fra | vedclass

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Question

$y = mx \pm \sqrt{ a ^{2} m ^{2}- b ^{2}}$

$m =2, c ^{2}= a ^{2} m ^{2}- b ^{2}$

$c ^{2}=4 a ^{2}- b ^{2}$

$e ^{2}=1+\frac{ b ^{2}}{ a ^{2}}$

Vedclass · Accepted Answer

$20$

Vedclass · Answer

$y = mx \pm \sqrt{ a ^{2} m ^{2}- b ^{2}}$

$m =2, c ^{2}= a ^{2} m ^{2}- b ^{2}$

$c ^{2}=4 a ^{2}- b ^{2}$

$e ^{2}=1+\frac{ b ^{2}}{ a ^{2}}$

$\frac{5}{2}=1+\frac{b^{2}}{a^{2}}$

$\frac{3}{2}=\frac{b^{2}}{a^{2}} \Rightarrow b^{2}=\frac{3 a^{2}}{2}$

$\frac{2 b^{2}}{a}=6 \sqrt{2}$

$\frac{2}{a} 	imes \frac{3 a^{2}}{2}=6 \sqrt{2}$

$3 a=6 \sqrt{2}$

$a=2 \sqrt{2}$

$b^{2}=\frac{3}{2} 	imes 8=12$

$b=2 \sqrt{3}$

$	herefore c^{2}=4 	imes 8-12$

$c^{2}=20$

Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y =2 x + c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

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