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10-2. Parabola, Ellipse, Hyperbola
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The line $2 \mathrm{x}+\mathrm{y}=1$ is tangent to the hyperbola $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$. If this line passes through the point of intersection of the nearest directrix and the $\mathrm{x}$-axis, then the eccentricity of the hyperbola is
A
$1$
B
$2$
C
$3$
D
$4$
(IIT-2010)
Solution
Substituting $\left(\frac{a}{e}, 0\right)$ in $y=-2 x+1$
$0=-\frac{2 a}{e}+1$
$\frac{2 a}{e}=1 $
$a=\frac{e}{2}$
Also, $1=\sqrt{\mathrm{a}^2 \mathrm{~m}^2-\mathrm{b}^2}$
$1=a^2 m^2-b^2$
$1=4 a^2-b^2$
$1=\frac{4 e^2}{4}-b^2$
$\mathrm{b}^2=\mathrm{e}^2-1 \text {. }$
Also, $\mathrm{b}^2=\mathrm{a}^2\left(\mathrm{e}^2-1\right)$
$\therefore \mathrm{a}=1, \mathrm{e}=2$
Standard 11
Mathematics
