Question

If a variable line has its intercepts on the co-ordinates 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. $2$
• B. $1$
• C. $3$
• D. Cannot be decided

Answer 1

Answer: (A)

$2$

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

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

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

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

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

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

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

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

$\therefore r=2$