The line 2x + 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

Engineering

Mathematics

Tangent to Hyperbola

Question

The line 2x + 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

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y = – 2x + 1

1 = 4a² – b²

4a² – b² = 1 passes (a/e, 0)

$\frac{2 a}{3} = 1 \Rightarrow$ e = 2a

$e^{2} = 1 + \frac{b^{2}}{a^{2}} = 1 + \frac{4 (e^{2} - 1)}{e^{2}}$

$e^{2} - 1 = \frac{4 (e^{2} - 1)}{e^{2}} \Rightarrow e = 2$