Question

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
$2x+3y+4z-12=0$

Answer 1

Direction ratio of any line normal to

$2x+3y+4z-12=0$ is $(2,3,4)$

$\therefore$ Equation of line through $(O,O,O)$ and is ${}^{\prime }{\perp }^{\prime }$ to given lines is

$\frac{x-0}{(}2)=\frac{y-0}{=}\frac{z-4}{4}$

any point on the lines is $(2\lambda ,3\lambda ,4\lambda )$ but this lines on the given plane

$2\left(2\lambda \right)+3\left(3\lambda \right)+4\left(4\lambda \right)-12=0\Rightarrow \lambda =\frac{12}{29}$

$\therefore$ Food of $\perp$ is $(2\times \frac{12}{29},3\times \frac{12}{29},4\times \frac{12}{29})$

$=(\frac{24}{29},\frac{36}{29},\frac{48}{29})$.