JEE Main 2020 (Online) 5th September Evening ShiftQuestion 1
If the line $y=mx+c$ is a common tangent to the hyperbola $\dfrac{x^{2}}{100}-\dfrac{y^{2}}{64}=1$ and the circle $x^{2}+y^{2}=36$, then which one of the following is true?
Hyperbola
Solution
Tangency to the circle: $c^2=36(1+m^2)$; to the hyperbola: $c^2=100m^2-64$. So $64m^2=100$, $m^2=\frac{25}{16}$ and $c^2=36\cdot\frac{41}{16}=\frac{369}4$, i.e. $4c^2=369$.