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10-2. Parabola, Ellipse, Hyperbola
hard
Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^{2} x^{2}-y^{2}=b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2}-\left(\frac{\mu}{b}\right)^{2}$ is equal to
A
$-2$
B
$-4$
C
$2$
D
$4$
(JEE MAIN-2022)
Solution
$\lambda x -2 y =\mu$ is a tangent to the curve
$a^{2} x^{2}-y^{2}=b^{2}$ then
$a ^{2} x ^{2}-\left(\frac{\lambda x -\mu}{2}\right)^{2}= b ^{2}$
$\left(4 a ^{2}-\lambda^{2}\right) x ^{2}+2 \lambda \mu x -\mu^{2}-4 b ^{2}=0$
Disc. $=0$
$4 \lambda^{2} \mu^{2}+4\left(4 a ^{2}-\lambda^{2}\right)\left(\mu^{2}+4 b ^{2}\right)=0$
$4 \lambda^{2} b^{2}-4 a^{2} \mu^{2}=16 a^{2} b^{2}$
$\frac{\lambda^{2}}{a^{2}}-\frac{\mu^{2}}{b^{2}}=4$
Standard 11
Mathematics
