The line 2 x + y = 1 is tangent to the hyperbola x 2 a 2 - 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]

Question

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]

Smart Achievers · Accepted Answer

(2)

Intersection point of nearest directrix $x = \frac{a}{e}$ and $x$ -axis is $(\frac{a}{e}, 0)$

Since $2 x + y = 1$ passes through $(\frac{a}{e}, 0)$ ,

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

Also $y = - 2 x + 1$ is a tangent to $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

$∴$ $1 = a^{2} {(- 2)}^{2} - b^{2}$ $\Rightarrow 4 a^{2} - b^{2} = 1$

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

$\Rightarrow e^{2} = 4$ as $e > 1$ for hyperbola, $\Rightarrow e = 2$

Solution

Q.
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

Q. 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]

Ans. (2) Intersection point of nearest directrix x=ae and x-axis is (ae,0) Since 2x+y=1 passes through (ae,0), ∴ 2ae=1⇒a=e2 Also y=-2x+1 is a tangent to x2a2-y2b2=1 ∴ 1=a2(-2)2-b2⇒4a2-b2=1 ⇒4a2-a2(e2-1)=1⇒4×e24-e24(e2-1)=1 ⇒e2=4 as e>1 for hyperbola, ⇒e=2

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Q.
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]