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

2

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1

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\sqrt{2}

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4

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Solution

The correct option is A $2$
Equation of tangent is $2 x + y = 1$
Given equation of hyberbola,
$\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$
Condition for tangency
$c^{2} = a^{2} m^{2} - b^{2}$
$1 = 4 a^{2} - b^{2} . . . . . (1)$
Line passes through $(\frac{- a}{e}, 0)$
So, $2 (- \frac{a}{e}) = 1$ .
$4 a^{2} = e^{2} . . . . . (2)$
From (1) and (2), $e = 2$ .

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