Question

Find the co-ordinates of the foot of the perpendicular drawn from the point$(2,3)$to the line $y=3x+4$

• A. $\left(\frac{1}{10},\frac{37}{10}\right)$
• B. $\left(\frac{1}{10},\frac{-37}{10}\right)$
• C. $\left(\frac{-1}{10},\frac{-37}{10}\right)$
• D. $\left(\frac{-1}{10},\frac{37}{10}\right)$

Answer 1

The correct option is D $\left(\frac{-1}{10},\frac{37}{10}\right)$

Equation of given line is $y=3x+4$
Let $Q(x_1,y_1)$ be the foot of the perpendicular on the line.
$\Rightarrow y_1=3x_1+4$ ......(i)
Slope of given line is $3.$
Slope of perpendicular $PQ=-\frac{1}{3}$
$\Rightarrow \frac{y_1-3}{x_1-2}=-\frac{1}{3}$
$\Rightarrow 3y_1=-x_1+11$ .....(ii)
Solving (i) and (ii), we get
$x_1=-\frac{1}{10},y_1=\frac{37}{10}$