Question

The line $2x+y=1$ is a 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 directrix and $x-$axis, then eccentricity of hyperbola is

Answer 1

For the line $y=-2x+1$ to be tangent to hyperbola,
$1=a^2(-2{)}^2-b^2\Rightarrow 4a^2-a^2(e^2-1)=1\Rightarrow e^2=\frac{5a^2-1}{a^2}\cdots (1)$
Also, tangent line passes through $(\frac{a}{e},0)$
So, $2a=e$
$\Rightarrow a=\frac{e}{2}\cdots \left(2\right)$
Solving $\left(1\right)$ and $\left(2\right)$
we get $e=2,e=1$
$\therefore e=2(\because e>1)$