The line 2 x+y=1 is a tangent to the hyperbola x^2 a^2 - y^2 b^2 =1(a>b) . If this line passes through the point of intersection of a directrix and the positive X -axis, then the eccentricity of that hyperbola is

Step-by-step solution

The line 2 x+y-1=0 is a tangent to the hyperbola x^2 a^2 - y^2 b^2 =1 and also passes through the point of intersection of directrix and the positive x -axis Equation of direction x= a e Point ( a e , 0 ) passes through 2 x+y=1 2 ( a e )+0=1 2 a=e 2 a^2=a e We know, (a e)^2=a^2+b^2 4 a^4=a^2+b^2 Equation of tangent of x^2 a^2 - y^2 b^2 =1 is y=m x a^2 m^2-b^2 a^2(2)^2-b^2=1 4 a^2-b^2=1 From Eqs. (i) and (ii), we get array ll & 4 a^4=a^2+4 a^2-1 & 4 a^4-5 a^2+1=0 & (4 a^2-1 ) (a^2-1 )=0 & a^2=1, 1 4 a=1, 1 2 & e=2 a