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The line 2x + y = 1 is tangent to the h...

The line `2x + 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

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The correct Answer is:
B
Substituting `(a//e,0)` in `y=-2x+1`, we get
`0=-(2a)/(e)+1`
`"or "(2a)/(e)=1`
`"or "a=(e)/(2)`
Also, `1=sqrt(a^(2)m^(2)-b^(2))`
`"or "1=4a^(2)-b^(2)`
`"or "1=(4e^(2))/(4)-b^(2)`
`"or "b^(2)=e^(2)-1`
Also, `b^(2)=a^(2)(e^(2)-1)`
`therefore" "a=1,e=2`
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