The coordinates of the foot of the perpendicular from the point...

Let $(h, k)$ be the coordinates of the foot of the perpendicular from the point $(2,3)$ on the $\mathrm{i}x+y-11=0$. Then, the slope of the perpendicular line is $\frac{k-3}{h-2}$. Again the slope of the given line $x+y-11=0$ is -1
Using the condition of perpendicularity of lines, we have


Since $(h, k)$ lies on the given line, we have,

Solving (i) and (ii), we get $h=5$ and $k=6$. Thus $(5,6)$ are the required coordinates of the foot of the perpendicular.