Question

The co-ordinates of the foot of the perpendicular from the point \( (0, 2, 3) \) on the line
\[ \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} \]

• A. \( (-2, -3, 1) \)
• B. \( (2, 3, -1) \)
• C. \( (2, 3, 1) \)
• D. \( (-2, -3, -1) \)

Hint:

For problems involving perpendiculars from points to lines, use the parametric equations of the line to substitute the coordinates of the point.

Answer 1

The Correct Option is B

Step 1: Parametric equations of the line.
The given equation of the line is: \[ \frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3} = t \] Thus, the parametric equations are: \[ x = 5t - 3, \quad y = 2t + 1, \quad z = 3t - 4 \] Step 2: Equation of the line joining the point to the foot of the perpendicular.
The point is \( P(0, 2, 3) \), and the direction ratios of the line are \( 5, 2, 3 \). The foot of the perpendicular \( Q(x_1, y_1, z_1) \) lies on the line. By substituting the coordinates of the point into the equation of the line, we get the coordinates of the foot of the perpendicular as \( (2, 3, -1) \). Step 3: Conclusion.
Thus, the co-ordinates of the foot of the perpendicular are \( \boxed{(2, 3, -1)} \).