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Standard XII

Mathematics

Position of a Line W.R.T Hyperbola

The line 2 x ...

Question

The line $2 x + y = 1$ 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

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Solution

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

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Position of a Line W.R.T Hyperbola

Standard XII Mathematics

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