Question

The coordinates of the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$ is

• A. $(-6,5)$
• B. $(5,6)$
• C. $(-5,6)$
• D. $(6,5)$

Answer 1

Answer: (B)

Let $(h,k)$ be the coordinates of the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$.



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
$\left(\frac{k-3}{h-2}\right)\left(-1\right)=-1$
$\Rightarrow k-h=1\dots \left(i\right)$
Since $\left(h,k\right)$ lies on the given line, we have,
$h+k-11=0$
$\Rightarrow h+k=11\dots \left(ii\right)$
Solving $\left(i\right)$ and $\left(ii\right)$, we get $h=5$ and $k=6$. Thus $\left(5,6\right)$ is the required coordinates of the foot of the perpendicular.