Question

If $3x-4y+k=0$ is a tangent to $x^2-4y^2=5$, find the value of $k$.

Answer 1

Condition for the line $y=mx+c$ is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $c=\pm \sqrt{a^2{m}^2-b^2}$

Here $a^2=5,b^2=\frac{5}{4},m=\frac{3}{4},c=\frac{k}{4}$

$\Rightarrow c=\pm \sqrt{a^2{m}^2-b^2}$

$\Rightarrow \frac{k^2}{16}=\frac{5.9}{16}-\frac{5}{4}$

$\Rightarrow k^2=45-20=25$

$\Rightarrow k=\pm 5$