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Q.
If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{ x ^{2}}{100}-\frac{ y ^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36,$ then which one of the following is true?
$y = mx + c$ is tangent to
$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and $x^{2}+y^{2}=36$
$c^{2}=100 m ^{2}-64 \mid c ^{2}=36\left(1+ m ^{2}\right)$
$\Rightarrow 100 m ^{2}-64=36+36 m ^{2}$
$m ^{2}=\frac{100}{64} $
$\Rightarrow m =\pm \frac{10}{8}$
$c^{2}=36\left(1+\frac{100}{64}\right)=\frac{36 \times 164}{64}$
$4 c^{2}=369$