SATHEE: Hyperbola 2 Question 15

Hyperbola 2 Question 15

15. 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…… (2010)

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Solution:

On substituting $\frac{a}{e}, 0$ in $y = - 2 x + 1$ , we get

$\begin{aligned} 0 & = - \frac{2 a}{e} + 1 \\ \Rightarrow & \frac{a}{e} & = \frac{1}{2} \end{aligned}$

Also, $y = - 2 x + 1$ is tangent to hyperbola.

$\begin{aligned} ∴ 1 = 4 a^{2} - b^{2} \Rightarrow \frac{1}{a^{2}} = 4 - (e^{2} - 1) \\ \Rightarrow \frac{4}{e^{2}} = 5 - e^{2} \\ \Rightarrow e^{4} - 5 e^{2} + 4 = 0 \Rightarrow (e^{2} - 4) (e^{2} - 1) = 0 \\ \Rightarrow e = 2, e = 1 \end{aligned}$

$e = 1$ gives the conic as parabola. But conic is given as hyperbola, hence $e = 2$ .

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Hyperbola 2 Question 15

15. The line 2x+y=1 is tangent to the hyperbola x2a2−y2b2=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…… (2010)

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15. 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…… (2010)