Question

The coordinates of the foot of the perpendicular from the point $(2,4)$ on the line $x+y=1$ are

• A. $\left(\frac{1}{3},\frac{3}{2}\right)$
• B. $\left(-\frac{1}{2},\frac{3}{2}\right)$
• C. $\left(\frac{4}{3},\frac{1}{2}\right)$
• D. $\left(\frac{3}{4},-\frac{1}{2}\right)$

Answer 1

The correct option is B $\left(-\frac{1}{2},\frac{3}{2}\right)$

Equation of given line is $x+y=1$
Let $Q(x_1,y_1)$ be the foot of the perpendicular on the line.
$\Rightarrow y_1=-x_1+1$ ......(i)
Slope of given line is $-1.$
Slope of perpendicular line $PQ$ is $1$
$\Rightarrow \frac{y_1-4}{x_1-2}=1$
$\Rightarrow y_1=x_1+2$ .....(ii)
Solving (i) and (ii), we get
$x_1=-\frac{1}{2},y_1=\frac{3}{2}$