• # question_answer Let a variable line has its intercepts on the coordinate axes, respectively as e,e' where $\frac{e}{2}$and $\frac{e'}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle ${{x}^{2}}+{{y}^{2}}={{a}^{2}},$ where a is equal to A)  4                                 B)  3               C)  2                                 D)  1

Since, $\frac{e}{2}$ and $\frac{e'}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola. Then, $\frac{1}{{{\left( \frac{e}{2} \right)}^{2}}}+\frac{1}{{{\left( \frac{e'}{2} \right)}^{2}}}=1$ $\Rightarrow$            $1=\frac{4}{{{e}^{2}}}+\frac{4}{e{{'}^{2}}}$ $\Rightarrow$            $4=\frac{{{e}^{2}}e{{'}^{2}}}{{{e}^{2}}+{{e}^{'2}}}$             ?(i)