Two tangents to the hyperbola $\frac{x^{2}}{100} - \frac{y^{2}}{81} = 1$ having slopes $m_{1} and m_{2}$ cuts the co-ordinate axes at four concyclic points. If $m_{1} and m_{2}$ satisfy the equation $2 α^{2} - 5 α + k = 0,$ then the value of ‘K’ is

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Detailed Solution

Equation of tangents of the hyperbola $\frac{x^{2}}{100} - \frac{y^{2}}{81} = 1$ are
$y = m_{1} x + \sqrt{100 {m_{1}}^{2} - 81} a n d y = m_{2} x + \sqrt{100 {m_{2}}^{2} - 81}$
These tangents cuts the coordinate axes at four points .
They are
$(\frac{- \sqrt{100 {m_{1}}^{2} - 81}}{m_{1}}, 0), (0, \sqrt{100 {m_{1}}^{2} - 81}), (\frac{- \sqrt{100 {m_{2}}^{2} - 81}}{m_{2}}, 0), (0, \sqrt{100 {m_{2}}^{2} - 81})$
since these points are concyclic

$\Rightarrow (\frac{- \sqrt{100 {m_{1}}^{2} - 81}}{m_{1}}) (\frac{- \sqrt{100 {m_{2}}^{2} - 81}}{m_{2}}) = \sqrt{100 {m_{1}}^{2} - 81} . \sqrt{100 {m_{2}}^{2} - 81} \Rightarrow m_{1} m_{2} = 1 \Rightarrow \frac{k}{2} = 1$