Question

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

• A. 1
• B. 2
• C. 3
• D. cannot be found

Answer 1

Answer: (B)

Since $\frac{ e }{2}$ and $\frac{ e ^{\prime}}{2}$ are eccentricities of hyperbola and its conjugate hyperbola hence, $\frac{4}{ e ^2}+\frac{4}{ e ^{\prime 2}}=1 \ldots$ (1)
equation of the line is $\frac{x}{e}+\frac{y}{e^{\prime}}=1$
$e^{\prime} x+e y-e e^{\prime}=0$
which touches the circle $x^2+y^2=r^2$
Hence, $\frac{ ee ^{\prime}}{\sqrt{ e ^2+\left( e ^{\prime}\right)^2}}= r \Rightarrow r ^2=4$ or $r =2\{$ from $(1)\}$