The coordinates of foot of the perpendicular drawn from the origin to the plane $2 x-3 y+4 z=29$ are

• A

  $(2,3,4)$
• B

  $(2,-3,-4)$
• Correct answer
  C

  $(2,-3,4)$
• D

  $(-2,-3,4)$

Solution

$$ \text { Given, equation of plane is } 2 x-3 y+4 z=29 $$
Since, $O P$ is perpendicular to the plane $$ 2 x-3 y+4 z=29 $$ Therefore, Direction Ratios of $O P$ will be $\langle 2,-3,4\rangle$. Equation of line $O P$ will be $$ \frac{x-0}{2}=\frac{y-0}{-3}=\frac{z-0}{4}=\lambda(\text { say }) $$ Coordinates of $P$ will be $(2 \lambda,-3 \lambda, 4 \lambda)$. $\because P$ lies on plane, so $2(2 \lambda)-3(-3 \lambda)+4(4 \lambda)=29$ $\Rightarrow \quad 4 \lambda+9 \lambda+16 \lambda=29$ $\Rightarrow \quad \lambda=1$ $\therefore$ Coordinates of $P$ will be $(2,-3,4)$.

Hence, the correct answer is option: 'C'.