The perpendicular from the origin to a line meets it at the point (– 2, 9),
Question:

The perpendicular from the origin to a line meets it at the point (– 2, 9), find the equation of the line.

Solution:

The slope of the line joining the origin $(0,0)$ and point $(-2,9)$ is $m_{1}=\frac{9-0}{-2-0}=-\frac{9}{2}$

Accordingly, the slope of the line perpendicular to the line joining the origin and point (– 2, 9) is

$m_{2}=-\frac{1}{m_{1}}=-\frac{1}{\left(-\frac{9}{2}\right)}=\frac{2}{9}$

Now, the equation of the line passing through point $(-2,9)$ and having a slope $m_{2}$ is

$(y-9)=\frac{2}{9}(x+2)$

$9 y-81=2 x+4$

i.e., $2 x-9 y+85=0$