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10-2. Parabola, Ellipse, Hyperbola
hard
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?
A
$5 m =4$
B
$4 c^{2}=369$
C
$c^{2}=369$
D
$8 m+5=0$
(JEE MAIN-2020)
Solution
$y=m x+c$ is tangent to
$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$ and $x^{2}+y^{2}=36$
$c^{2}=100 m^{2}-64 l c^{2}=36\left(1+m^{2}\right)$
$\Rightarrow 100 m ^{2}-64=36+36 m ^{2}$
$m ^{2}=\frac{100}{64} \Rightarrow m =\pm \frac{10}{8}$
$c^{2}=36\left(1+\frac{100}{64}\right)=\frac{36 \times 164}{64}$
$4 c^{2}=369$
Standard 11
Mathematics
