A variable line having intercepts e and e prime on co-ordinate axes, where (e/2), (e prime/2) are eccentricities of a hyperbola and its conjugate hyperbola , then the line touches the circle x2+y2=r2 whose radius is

Question

A variable line having intercepts e and e prime on co-ordinate axes, where (e/2), (e prime/2) are eccentricities of a hyperbola and its conjugate hyperbola , then the line touches the circle x2+y2=r2 whose radius is (A) 1 (B) 2 (C) 3 (D) cannot be found

Tardigrade Admin · Accepted Answer

Since ( e /2) and ( e prime/2) are eccentricities of hyperbola and its conjugate hyperbola hence, (4/ e 2)+(4/ e prime 2)=1 ldots (1) equation of the line is (x/e)+(y/e prime)=1 e prime x+e y-e e prime=0 which touches the circle x2+y2=r2 Hence, ( ee prime/√ e 2+( e prime)2)= r ⇒ r 2=4 or r =2 from (1)

Q. A variable line having intercepts e and e′ on co-ordinate axes, where 2e​,2e′​ are eccentricities of a hyperbola and its conjugate hyperbola , then the line touches the circle x2+y2=r2 whose radius is

1

2

3

cannot be found

Since 2e​ and 2e′​ are eccentricities of hyperbola and its conjugate hyperbola hence, e24​+e′24​=1… (1) equation of the line is ex​+e′y​=1 e′x+ey−ee′=0 which touches the circle x2+y2=r2 Hence, e2+(e′)2​ee′​=r⇒r2=4 or r=2{ from (1)}

Q. A variable line having intercepts $e$ and $e^{'}$ on co-ordinate axes, where $\frac{e}{2}, \frac{e ^{'}}{2}$ are eccentricities of a hyperbola and its conjugate hyperbola , then the line touches the circle $x^{2} + y^{2} = r^{2}$ whose radius is