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
$$ \begin{array}{rlrl} 0 & =-\frac{2 a}{e}+1 \\ \Rightarrow \quad & \frac{a}{e} & =\frac{1}{2} \end{array} $$
Also, $y=-2 x+1$ is tangent to hyperbola.
$$ \begin{aligned} & \therefore \quad 1=4 a^{2}-b^{2} \Rightarrow \frac{1}{a^{2}}=4-\left(e^{2}-1\right) \\ & \Rightarrow \quad \frac{4}{e^{2}}=5-e^{2} \\ & \Rightarrow \quad e^{4}-5 e^{2}+4=0 \Rightarrow\left(e^{2}-4\right)\left(e^{2}-1\right)=0 \\ & \Rightarrow \quad e=2, e=1 \end{aligned} $$
$e=1$ gives the conic as parabola. But conic is given as hyperbola, hence $e=2$.
