Question:
If the foot of the perpendicular drawn from the origin to a plane is \( P(2, -1, 4) \), then the equation of the plane is
If the foot of the perpendicular drawn from the origin to a plane is \( P(2, -1, 4) \), then the equation of the plane is
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If origin is $(0,0,0)$ and foot is $(x_1, y_1, z_1)$, the plane is simply $x_1x + y_1y + z_1z = x_1^2 + y_1^2 + z_1^2$.
Updated On: Apr 30, 2026
- \( 2x + y + 4z - 19 = 0 \)
- \( x + y + z - 5 = 0 \)
- \( 2x - 2y - 3z + 6 = 0 \)
- \( 2x - y + 4z - 21 = 0 \)
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The Correct Option is D
Solution and Explanation
Step 1: Find Normal Vector
The line joining the origin (0,0,0) and the foot $P(2, -1, 4)$ is the normal to the plane. Normal DRs: $(2-0, -1-0, 4-0) = (2, -1, 4)$.
Step 2: General Plane Equation
$a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$. $2(x-2) - 1(y+1) + 4(z-4) = 0$.
Step 3: Simplification
$2x - 4 - y - 1 + 4z - 16 = 0$. $2x - y + 4z - 21 = 0$.
Step 4: Conclusion
The equation is $2x - y + 4z - 21 = 0$.
Final Answer:(D)
The line joining the origin (0,0,0) and the foot $P(2, -1, 4)$ is the normal to the plane. Normal DRs: $(2-0, -1-0, 4-0) = (2, -1, 4)$.
Step 2: General Plane Equation
$a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$. $2(x-2) - 1(y+1) + 4(z-4) = 0$.
Step 3: Simplification
$2x - 4 - y - 1 + 4z - 16 = 0$. $2x - y + 4z - 21 = 0$.
Step 4: Conclusion
The equation is $2x - y + 4z - 21 = 0$.
Final Answer:(D)
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