Question

The equation of the perpendicular line from the point \((2,-3,1)\) to the line \(\frac{x+1}{2} = \frac{y-3}{3} = \frac{z+2}{-1}\) is

• A. \(\frac{x-2}{24} = \frac{y+3}{13} = \frac{z-1}{9}\)
• B. \(\frac{x-2}{24} = \frac{y-3}{-13} = \frac{z-1}{9}\)
• C. \(\frac{x+2}{-24} = \frac{y+3}{13} = \frac{z+1}{-9}\)
• D. \(\frac{x-2}{-24} = \frac{y+3}{13} = \frac{z-1}{-9}\)

Hint:

On the YOZ plane x = 0, so find the parameter where the x coordinate vanishes.

Answer 1

The Correct Option is D

Step 1: Section formula:
The YOZ plane divides the segment in the ratio $\lambda : 1$ where $\frac{-2\lambda + 3}{\lambda + 1} = 0$, so $\lambda = \frac32$.

Step 2: Compute:
$y = \frac{1\cdot\frac32 + 5}{\frac32 + 1} = \frac{6.5}{2.5} = \frac{13}{5}$ and $z = \frac{8\cdot\frac32 - 7}{2.5} = \frac{5}{2.5} = 2$. Option (C).

Final Answer:
$\left(0,\frac{13}{5},2\right)$. \[ \boxed{\left(0,\frac{13}{5},2\right)} \]