A plane makes intercepts a,b,c at A,B,C on the coordinate axes respectively. If the centroid of the `DeltaABC` is at (3,2,1), then the equation of the plane is

Options

A. `x+2y+3z=9`
B. `2x-3y-6z=18`
C. `2x+3y+6z=18`
D. `2x+y+6z=18`

Explanation

To find the equation of the plane that makes intercepts \( a, b, c \) at points \( A, B, C \) on the coordinate axes, given that the centroid of triangle \( ABC \) is at \( (3, 2, 1) \), we can follow these steps:

### Step 1: Identify the coordinates of points A, B, and C
The points where the plane intersects the coordinate axes are:
- Point \( A \) (x-intercept): \( (a, 0, 0) \)
- Point \( B \) (y-intercept): \( (0, b, 0) \)
- Point \( C \) (z-intercept): \( (0, 0, c) \)

### Step 2: Write the equation of the plane in intercept form
The equation of the plane in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]

### Step 3: Use the centroid formula
The coordinates of the centroid \( G \) of triangle \( ABC \) are given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]
Substituting the coordinates of points \( A, B, C \):
\[ G = \left( \frac{a + 0 + 0}{3}, \frac{0 + b + 0}{3}, \frac{0 + 0 + c}{3} \right) = \left( \frac{a}{3}, \frac{b}{3}, \frac{c}{3} \right) \]
We know the centroid is at \( (3, 2, 1) \), so we can set up the following equations:
\[ \frac{a}{3} = 3 \quad (1) \]
\[ \frac{b}{3} = 2 \quad (2) \]
\[ \frac{c}{3} = 1 \quad (3) \]

### Step 4: Solve for a, b, and c
From equation (1):
\[ a = 3 \times 3 = 9 \]
From equation (2):
\[ b = 3 \times 2 = 6 \]
From equation (3):
\[ c = 3 \times 1 = 3 \]

### Step 5: Substitute a, b, and c into the plane equation
Now we substitute \( a = 9 \), \( b = 6 \), and \( c = 3 \) into the plane equation:
\[ \frac{x}{9} + \frac{y}{6} + \frac{z}{3} = 1 \]

### Step 6: Clear the denominators
To eliminate the fractions, multiply the entire equation by the least common multiple of the denominators (which is 18):
\[ 18 \left( \frac{x}{9} + \frac{y}{6} + \frac{z}{3} \right) = 18 \]
This simplifies to:
\[ 2x + 3y + 6z = 18 \]

### Final Equation of the Plane
Thus, the equation of the plane is:
\[ 2x + 3y + 6z = 18 \]