Question:
If the line $y=m x+c$ 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?
Correct Option: , 3
Solution:
General tangent to hyperbola in slope form is
$y=m x \pm \sqrt{100 m^{2}-64}$
and the general tangent to the circle in slope form is
$y=m x \pm 6 \sqrt{1+m^{2}}$
For common tangent,
$36\left(1+m^{2}\right)=100 m^{2}-64$
$\Rightarrow 100=64 m^{2} \Rightarrow m^{2}=\frac{100}{64}$
$\therefore c^{2}=36\left(1+\frac{100}{64}\right)=\frac{164 \times 36}{64}=\frac{369}{4}$
$\Rightarrow 4 c^{2}=369$