Question

Find the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4).

Answer 1

Let A (−1, 2) be the given point whose projection is to be evaluated and C (−1, 2) and D (5, 4) be the other two points.

Also, let M (h, k) be the foot of the perpendicular drawn from A (−1, 2) to the line joining the points C (−1, 2) and D (5, 4).



Clearly, the slope of CD and MD are equal.

$\therefore \frac{4-k}{5-h}=\frac{4-2}{5+1}$

$\Rightarrow h-3k+7=0$ ... (1)

The lines segments AM and CD are perpendicular.

$\therefore$ $\frac{k-0}{h-1}\times \frac{4-2}{5+1}=-1$

$\Rightarrow 3h+k-3=0$ ... (2)

Solving (1) and (2) by cross multiplication, we get:

$$\begin{gathered} \frac{h}{9-7}=\frac{k}{21+3}=\frac{1}{1+9} \\ \Rightarrow h=\frac{1}{5},k=\frac{12}{5} \end{gathered}$$

Hence, the projection of the point (1, 0) on the line joining the points (−1, 2) and (5, 4) is $\left(\frac{1}{5},\frac{12}{5}\right)$.