JEE Main 2022 (Online) 24th June Morning ShiftQuestion 1
Let $\lambda x-2y=\mu$ 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 :
Hyperbola
Solution
Write the hyperbola as $\dfrac{x^2}{b^2/a^2}-\dfrac{y^2}{b^2}=1$. The line $y=\dfrac\lambda2x-\dfrac\mu2$ is tangent when $c^2=A^2m^2-B^2$: $\dfrac{\mu^2}{4}=\dfrac{b^2}{a^2}\cdot\dfrac{\lambda^2}{4}-b^2$. Multiply by $\dfrac{4}{b^2}$: $\dfrac{\mu^2}{b^2}=\dfrac{\lambda^2}{a^2}-4\Rightarrow\left(\dfrac\lambda a\right)^2-\left(\dfrac\mu b\right)^2=4$.