Question

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

• 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
$\left(\frac{k-3}{h-2}\right)(-1)=-1\left(\because m_1{m}_2=-1\right)$
or $k-h=\mathrm{1....}$(i)
Since, $(h,k)$ lies on the given line, so we have
$h+k-11=0$
or $h+k=\mathrm{11....}$(ii)
On solving Eqs. (i) and (ii), we get $h=5$ and $k=6$
Thus, $(5,6)$ are the required coordinates of the foot of perpendicular.