The line $2 x+y=1$ is tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. If this

Question

The line $2 x+y=1$ is tangent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. If this line passes through the point of intersection of the nearest directrix and the $x$-axis, then the eccentricity of the hyperbola is

Professor PC Thomas Classes · Answer

(2) : $y=-2 x+1$ passes through nearest directrix\ $\Rightarrow 0=\frac{-2 a}{e}+1 \Rightarrow \frac{2 a}{e}=1 \Rightarrow \frac{a}{e}=\frac{1}{2} \Rightarrow e=2 a$\ Now $e^{2}=1+\frac{b^{2}}{a^{2}} \Rightarrow e^{2}=5-\frac{4}{e^{2}}$\ $\Rightarrow e^{4}-5 e^{2}+4=0$\ $\Rightarrow \quad\left(e^{2}-1ight)\left(e^{2}-4ight)=0 	herefore \quad e^{2}=4 \Rightarrow e=2$\