Question

Foot of the perpendicular drawn from the origin to the plane \( 2x - 3y + 4z = 29 \) is:

• A. \( (5, -1, 4) \)
• B. \( (7, -1, 3) \)
• C. \( (5, -2, 3) \)
• D. \( (2, -3, 4) \)
• E. \( (1, -3, 4) \)

Hint:

For the foot of the perpendicular from the origin, the coordinates will always be in the ratio of the direction ratios of the normal ($A:B:C$). Here, $2:-3:4$ immediately points to Option (D).

Answer 1

The Correct Option is D

Concept: The foot of the perpendicular \( (x_1, y_1, z_1) \) from a point \( (x_0, y_0, z_0) \) to a plane \( Ax + By + Cz + D = 0 \) is given by the formula: \[ \frac{x_1 - x_0}{A} = \frac{y_1 - y_0}{B} = \frac{z_1 - z_0}{C} = -\frac{Ax_0 + By_0 + Cz_0 + D}{A^2 + B^2 + C^2} \]

Step 1: Substitute origin coordinates and plane parameters.
Origin \( (x_0, y_0, z_0) = (0, 0, 0) \). Plane: \( 2x - 3y + 4z - 29 = 0 \). So, \( A=2, B=-3, C=4, D=-29 \). \[ \frac{x_1}{2} = \frac{y_1}{-3} = \frac{z_1}{4} = -\frac{2(0) - 3(0) + 4(0) - 29}{2^2 + (-3)^2 + 4^2} \]

Step 2: Solve for the constant ratio.
\[ \text{Ratio} = -\frac{-29}{4 + 9 + 16} = \frac{29}{29} = 1 \]

Step 3: Calculate coordinates.
\[ x_1 = 2(1) = 2 \] \[ y_1 = -3(1) = -3 \] \[ z_1 = 4(1) = 4 \] The foot of the perpendicular is \( (2, -3, 4) \).