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  • The line is tangent to the hyperbola <img alt="\mathrm{\frac{x^2}{a^2}-\frac

The line \mathrm{2 x+y=1} is tangent to the hyperbola \mathrm{\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 ___________.

Option: 1

2


Option: 2

0


Option: 3

3


Option: 4

1


Answers (1)

best_answer

Point \left(\frac{a}{e}, 0\right) lies on line y=-2 x+1

\mathrm{ \Rightarrow 0=-\frac{2 a}{e}+1 \Rightarrow \frac{a}{e}=\frac{1}{2} \Rightarrow e=2 a \text {. } }

Condition of tangency; c^2=a^2 m^2-b^2

\mathrm{ \begin{aligned} & \Rightarrow \quad 1=4 a^2-b^2 \Rightarrow 1+b^2-4 a^2=0 \, \, \, \, ....(i)\\\\ & \because \quad e^2=1+\frac{b^2}{a^2}=1+\frac{\left(4 a^2-1\right)}{a^2} \end{aligned} }[From (i)]

\mathrm{\begin{aligned} & \Rightarrow e^2=1+4-\frac{1}{a^2} \\\\ & \Rightarrow e^2=5-\frac{4}{e^2} \\\\ & \Rightarrow e^4-5 e^2+4=0 \\\\ & \Rightarrow\left(e^2-1\right)\left(e^2-4\right)=0 \\\\ & \Rightarrow e^2-1 \neq 0, e=2 . \end{aligned}}

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Deependra Verma

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