A plane meets the co-ordinate axes in A,B,C, such that the centroid of `triangleABC` is (3,3,3). Then the equation of the plane is

Options

A. x+y+z=3
B. x+y+z=9
C. 3x+3y +3z=1
D. 9x+9y + 9z=1

Answer

To find the equation of the plane that meets the coordinate axes at points A, B, and C such that the centroid of triangle ABC is (3, 3, 3), we can follow these steps:

### Step 1: Define the Points A, B, and C
Let the coordinates of the points where the plane intersects the axes be:
- A = (a, 0, 0) (x-intercept)
- B = (0, b, 0) (y-intercept)
- C = (0, 0, c) (z-intercept)

### Step 2: Find the Centroid of Triangle ABC
The centroid \( G \) of triangle ABC is given by the formula:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]
Substituting the coordinates of points A, B, and C:
\[ G = \left( \frac{a + 0 + 0}{3}, \frac{0 + b + 0}{3}, \frac{0 + 0 + c}{3} \right) = \left( \frac{a}{3}, \frac{b}{3}, \frac{c}{3} \right) \]

### Step 3: Set the Centroid Equal to (3, 3, 3)
We know from the problem statement that the centroid is (3, 3, 3). Therefore, we can set up the following equations:
\[ \frac{a}{3} = 3 \quad \Rightarrow \quad a = 9 \]
\[ \frac{b}{3} = 3 \quad \Rightarrow \quad b = 9 \]
\[ \frac{c}{3} = 3 \quad \Rightarrow \quad c = 9 \]

### Step 4: Write the Equation of the Plane
The intercept form of the equation of a plane is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
Substituting the values of a, b, and c:
\[ \frac{x}{9} + \frac{y}{9} + \frac{z}{9} = 1 \]

### Step 5: Simplify the Equation
Multiplying through by 9 to eliminate the denominators:
\[ x + y + z = 9 \]

### Final Answer
Thus, the equation of the plane is:
\[ \boxed{x + y + z = 9} \]