Question

If a plane meets the axes $X$, $Y$, $Z$ in $A$, $B$, $C$ respectively such that centroid of $\Delta ABC$ is $(1, 2, 3)$, then the equation of the plane is

• A. $x + 2y + 3z = 1$
• B. $x + \frac{y}{2} + \frac{z}{3} = 3$
• C. $\frac{x}{3} + \frac{y}{6} + \frac{z}{9} = 1$
• D. $\frac{x}{4} + \frac{y}{8} + \frac{z}{12} = 1$

Hint:

For any plane whose axes-intercept triangle has a centroid $(x_0, y_0, z_0)$, the values of the intercepts are always exactly three times the centroid's coordinates ($a = 3x_0, b = 3y_0, c = 3z_0$). This allows you to write down the final equation $\frac{x}{3x_0} + \frac{y}{3y_0} + \frac{z}{3z_0} = 1$ in a single step!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
A plane meets axes at A, B, C. The centroid of ΔABC is (1, 2, 3). Find the plane's equation.

Step 2: Key Formula or Approach:
Let intercepts be a, b, c. Vertices are (a,0,0), (0,b,0), (0,0,c). Centroid = (a/3, b/3, c/3). Plane equation: x/a + y/b + z/c = 1.

Step 3: Detailed Explanation:
Equating: a/3 = 1 → a = 3; b/3 = 2 → b = 6; c/3 = 3 → c = 9. Substituting intercepts: x/3 + y/6 + z/9 = 1.

Step 4: Final Answer:
The plane equation is x/3 + y/6 + z/9 = 1, option (C).