Question

If a variable line has its intercepts on the coordinate axes $e,e^{\prime }$, where $\frac{e}{2},\frac{e^{\prime }}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle $x^2+y^2=r^2$, where $r=$

• A. $1$
• B. $2$
• C. $3$
• D. none of these

Answer 1

The correct option is B $2$

Since, $\frac{e}{2}$ and $\frac{e^{\prime }}{2}$ are eccentricities of a hyperbola and its conjugate hyperbola, therefore

$\frac{4}{e^2}+\frac{4}{e^{\prime 2}}=1$

$\Rightarrow 4=\frac{e^2{e}^{\prime 2}}{e^{\prime 2}+e^{\prime 2}}$

The line passing through the points $(e,0)$ and $(0,e^{\prime })$ is

$e^{\prime }x+ey-e{e}^{\prime }=0$

It is tangent to the circle $x^2+y^2=r^2$.

Hence, $\frac{e{e}^{\prime }}{\sqrt{e^2+e^{\prime 2}}}=r$

$\Rightarrow r^2=\frac{e^2{e}^{\prime 2}}{e^2+e^{\prime 2}}=4$

$\Rightarrow r=2$