Question

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?

• A. $5m=4$
• B. $4c^2=369$
• C. $c^2=369$
• D. $8m+5=0$

Answer 1

Answer: (B)

$y=mx+c$ is tangent to
$\frac{x^2}{100}-\frac{y^2}{64}=1$ and $x^2+y^2=36$
$c^2=100m^2-64\mid c^2=36\left(1+m^2\right)$
$\Rightarrow 100m^2-64=36+36m^2$
$m^2=\frac{100}{64}$
$\Rightarrow m=\pm \frac{10}{8}$
$c^2=36\left(1+\frac{100}{64}\right)=\frac{36\times 164}{64}$
$4c^2=369$