The line 2 x+y=1 is a tangent to the hyperbola...

The line $2 x+y-1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and also passes through the point of intersection of directrix and the positive $x$-axis Equation of direction $x=\frac{a}{e}\therefore$ Point $\left(\frac{a}{e}, 0\right)$ passes through $2 x+y=1\Rightarrow \quad 2\left(\frac{a}{e}\right)+0=1\Rightarrow \quad 2 a=e\Rightarrow \quad 2 a^2=a e$ We know, $(a e)^2=a^2+b^2\therefore \quad 4 a^4=a^2+b^2$ Equation of tangent of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $y=m x \pm \sqrt{a^2 m^2-b^2}\therefore \quad a^2(2)^2-b^2=1$ \Rightarrow \quad 4 a²-b²=1 From Eqs. (i) and (ii), we get $$\begin{array}{ll}& 4 a⁴=a²+4 a²-1 \\ \Rightarrow & 4 a⁴-5 a²+1=0 \\ & \left(4 a²-1\right)\left(a^2-1\right)=0 \\ \Rightarrow & a²=1, \frac{1}{4} \Rightarrowa=1, \frac{1}{2}\\ \therefore & e=2 a \\ & e=2(1)=2 \end{array}$$