Question

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

Answer 1

Hint: We start solving the problem by substituting the point $\left( -1,2 \right)$ in the line $y=mx+c$, as it must lie on it to get the relation between m and c. We then find the slope of the line passing through the points origin and $\left( -1,2 \right)$ using the fact that the slope of the line passing through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. We then find the value of m using the fact that the product of slopes of perpendicular lines is –1. We then substitute this value of m in the relation we obtained first to get the value of c.

Complete step-by-step solution:

According to the problem, we are given that the perpendicular from the origin to the line $y=mx+c$ meets it at the point $\left( -1,2 \right)$. We need to find the values of m and c.

Since the point $\left( -1,2 \right)$ is the foot of perpendicular from the origin to the line $y=mx+c$. This tells us the point $\left( -1,2 \right)$ lies on the line $y=mx+c$.

On substituting the point $\left( -1,2 \right)$ in the line $y=mx+c$, we get $2=-m+c$ -(1).

Let us find the slope of the line passing through the points origin $\left( 0,0 \right)$ and the point $\left( -1,2 \right)$.

We know that the slope of the line passing through the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ is $\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$.

So, the slope of line passing through the points origin $\left( 0,0 \right)$ and the point $\left( -1,2 \right)$ is $\dfrac{2-0}{-1-0}=\dfrac{2}{-1}=-2$. We know that this line is perpendicular to the line $y=mx+c$.

We know that the product of the slopes of two perpendicular lines is –1.

So, we get $m\times -2=-1$.

$\Rightarrow m=\dfrac{-1}{-2}=\dfrac{1}{2}$. Let us substitute this value in equation (1).

So, we get $2=-\dfrac{1}{2}+c$.

$\Rightarrow c=\dfrac{5}{2}$.

So, we have found the values of m and c as $\dfrac{1}{2}$ and $\dfrac{5}{2}$.

Note: We should know that the line passing through the foot of perpendicular and the point from which the foot is found will be perpendicular to the line that the foot of perpendicular is present which is a very important property. We should know that the foot of the perpendicular will always lie on that given line. We can also find the perpendicular distance from the origin to the lines using the points or the line we just obtained. Similarly, we can expect problems to find the equation of the line(s) parallel to the obtained line that is at a distance of 6 from the origin.