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 other 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.

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

$\Rightarrow h-3k+7=0...\left(i\right)$

The lines segments AM and CD are perpendicular.

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

$\Rightarrow 3h+k-3=0...\left(ii\right)$

Solving (i) and (ii) by cross multiplication, we get:

$\frac{c}{9-7}=\frac{k}{21+3}=\frac{1}{1+9}$

$\Rightarrow h=\frac{1}{5},k=\frac{12}{5}$

Hence, the projection of the point $(1,0)$ on the line joining the points $(-1,2)$ and

$(5,4)$ is $(\frac{1}{5},\frac{12}{5}).$