Question

For what value of $c$ will the line $y=2x+c$ be a tangent to the circle $x^2+y^2=5$

Answer 1

We have $y=2x+c$ or $2x-y+c=0$ ........(i)
$x^2+y^2=5$ ........(ii)
If the line (i) touches the circle (ii) then
length of the $\perp$ from the centre (0, 0) = radius of circle (ii)
$\Rightarrow \left|\frac{2\times 0-0+c}{\sqrt{2^2+{(-1)}^2}}\right|=\sqrt{5}\Rightarrow \left|\frac{c}{\sqrt{5}}\right|=\sqrt{5}$
$\Rightarrow \frac{c}{\sqrt{5}}=\pm \sqrt{5}$
$\Rightarrow c=\pm 5$
Hence the line (i) touches the circle (ii) for c = $\pm 5$