Question

Find the coordinates of the foot of the perpendicular from the point (−1, 3) to the line 3x − 4y − 16 = 0.

Answer 1

Let A (−1, 3) be the given point.
Also, let M (h, k) be the foot of the perpendicular drawn from A (−1, 3) to the line 3x − 4y − 16 = 0



Point M (h, k) lies on the line 3x − 4y − 16 = 0

3h − 4k − 16 = 0 ... (1)

Lines 3x − 4y − 16 = 0 and AM are perpendicular.

$\therefore$ $\frac{k-3}{h+1}\times \frac{3}{4}=-1$

$\Rightarrow 4h+3k-5=0$ ... (2)

Solving eq (1) and eq (2) by cross multiplication, we get:

$$\begin{gathered} \frac{h}{20+48}=\frac{k}{-64+15}=\frac{1}{9+16} \\ \Rightarrow a=\frac{68}{25},b=-\frac{49}{25} \end{gathered}$$

Hence, the coordinates of the foot of perpendicular are $\left(\frac{68}{25},-\frac{49}{25}\right)$.