Question

If the foot of the perpendicular drawn from the origin to a plane is P(-1, -1, 2), then the equation of the plane is ______.

• A. $x + y - 2z + 6 = 0$
• B. $2x + y + z + 1 = 0$
• C. $x + y + 2z - 2 = 0$
• D. $x - y - z + 2 = 0$

Hint:

Shortcut: If the foot of the perpendicular from the origin is $(x_1, y_1, z_1)$, the plane equation is incredibly simple: $x_1 x + y_1 y + z_1 z = x_1^2 + y_1^2 + z_1^2$.
Here: $-1x - 1y + 2z = 1 + 1 + 4 \implies -x - y + 2z = 6 \implies x + y - 2z + 6 = 0$. Extremely fast!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The vector $\vec{OP}$ (where $O$ is the origin) acts as the normal vector $\vec{n}$ to the plane. The plane passes through point $P$.

Step 2: Formula Application:
Normal vector $\vec{n} = -1\hat{i} - 1\hat{j} + 2\hat{k}$. Equation of plane: $a(x-x_1) + b(y-y_1) + c(z-z_1) = 0$.

Step 3: Explanation:
$-1(x - (-1)) - 1(y - (-1)) + 2(z - 2) = 0$ $-1(x + 1) - 1(y + 1) + 2z - 4 = 0$ $-x - 1 - y - 1 + 2z - 4 = 0 \implies -x - y + 2z - 6 = 0$. Multiplying by $-1$: $x + y - 2z + 6 = 0$.

Step 4: Final Answer:
The equation is $x + y - 2z + 6 = 0$.