The line 2 x+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

Step-by-step solution

On substituting ( a e , 0 ) in y=-2 x+1 , we get aligned 0 & =- 2 a e +1 a e & = 1 2 aligned Also, y=-2 x+1 is tangent to hyperbola array ll & 1=4 a^2-b^2 & 1 a^2 =4- (e^2-1 ) array array ll & 4 e^2 =5-e^2 & e^4-5 e^2+4=0 & (e^2-4 ) (e^2-1 )=0 & e=2, e=1 array e=1 gives the conic as parabola. But conic is given as hyperbola, hence e=2