If e₁ and e₂ are respectively the eccentricities of the hyperbola x^2 a^2 - y^2 b^2 =1 and its conjugate hyperbola, then the line x 2 e₁ + y 2 e₂ =1 touches the circle having centre at the origin, then its radius is

Step-by-step solution

Hyperbola : x^2 a^2 - y^2 b^2 =1 ; e₁= a^2+b^2 a^2 Conjugate Hyperbola : y^2 b^2 - x^2 a^2 =1 ; e₂= a^2+b^2 b^2 Line: x 2 e₁ + y 2 e₂ =1 a^2 x 2 (a^2+b^2 ) + b^2 y 2 (a^2+b^2 ) =1 a x+b y=2 (a^2+b^2 ) Since, the line touches the circle Distance from origin to line = radius 2 (a^2+b^2 ) a^2+b^2 =r r=2