Question

The perpendicular from the origin to the line $y$ $=mx+c$ meets it at the point $(-1,2)$. Find the values of $m$ and $c$.

Answer 1

The given equation of line is $y=mx+c$.
It is given that the perpendicular from the origin meets the given line at $(-1,2)$.

Therefore, the line joining the points( $0,0)$ and $(-1,2)$ is perpendicular to the given line.

$\therefore$ slope of the line joining $(0,0)$ and $(-1,2)$ is $=\frac{2}{-1}=-2$
The slope of the given line is $m$.
$\therefore m\times -2=-1$ [the two lines are perpendicular]
$\Rightarrow m=\frac{1}{2}$

Since point $(-1,2)$ lies on the given line, it satisfies the equation $y$ $=mx+c$
$\therefore 2=m(-1)+c$
$\Rightarrow 2=\frac{1}{2}(-1)+c$
$\Rightarrow c=2+\frac{1}{2}=\frac{5}{2}$
Thus, the respective values of $m$ and $c$ are $\frac{1}{2}$ and $\frac{5}{2}$.