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

Options

A. `(3)/(2)`
B. 2
C. `(5)/(2)`
D. 3

Answer

B

Explanation

y=-2x +1 passes through `((a)/(e),0)` ltBrgt `therefore(a)/(e)=(1)/(2)`
y = -2 x +1 touches the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`
`therefore 1=4a^(2)-b^(2)` ltBrgt `rArr 1=4a^(2)-b^(2)` ltBrgt `rArr 1=(e^(2))/(4)(5-e^(2))`
`rArre^(4)-5e^(2)+4=0`
`rArre^(2)=4,1` ltBrgt `thereforee^(2)ne1thereforee=2`