The line $$2x + y = 1$$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$.
If this line passes through the point of intersection of the nearest directrix and the $$x$$-axis, then the eccentricity of the hyperbola is
On substituting $$\left( {{a \over e},0} \right)$$ in $$y = - 2x + 1$$,
we get
$$0 = - {{2a} \over e} + 1$$
$$ \Rightarrow {a \over e} = {1 \over 2}$$
Also, $$y = - 2x + 1$$ is tangent to hyperbola
$$\therefore$$ $$1 = 4{a^2} - {b^2}$$
$$ \Rightarrow {1 \over {{a^2}}} = 4 - ({e^2} - 1)$$
$$ \Rightarrow {4 \over {{e^2}}} = 5 - {e^2}$$
$$ \Rightarrow {e^4} - 5{e^2} + 4 = 0$$
$$ \Rightarrow ({e^2} - 4)({e^2} - 1) = 0$$
$$\Rightarrow$$ e = 2, e = 1
e = 1 gives the conic as parabola. But conic is given as hyperbola, hence e = 2.
