Question

Find the equation of a line which slope $2$ and the length of the perpendicular from the origin is $\sqrt{5}$.

Answer 1

Given the slope of the line, $m=2$

and the perpendicular distance from the origin is $\sqrt{5}=d_{origin}$

The equation of a line with slope $m$ and y-intercept is,

$y=mx+C$

$\therefore$ The equation of the above line can be written as

$y=2x+C$

$\Rightarrow 2x-y+c=0$

Now, $d_{origin}=\sqrt{5}$

$\Rightarrow \frac{\mid Ax+By+C\mid }{\sqrt{A^2+B^2}}=\sqrt{5}$

$\Rightarrow \frac{C}{\sqrt{4}+1}=\sqrt{5}$

$\Rightarrow C=5$

$\therefore$ The line equation is $2x-y+5=0$