A plane intersects three co-ordinate axes at the points A, B, C. If (1, -2, 3) is the centroid of the`DeltaABC` then find the equation of the plane.

Answer

To find the equation of the plane that intersects the coordinate axes at points A, B, and C, given that the centroid of triangle ABC is (1, -2, 3), we can follow these steps:

### Step 1: Define the points of intersection
Let the points where the plane intersects the coordinate axes be:
- A = (x, 0, 0) (intersection with the x-axis)
- B = (0, y, 0) (intersection with the y-axis)
- C = (0, 0, z) (intersection with the z-axis)

### Step 2: Write the formula for the centroid
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, we have:
\[ G = \left( \frac{x + 0 + 0}{3}, \frac{0 + y + 0}{3}, \frac{0 + 0 + z}{3} \right) = \left( \frac{x}{3}, \frac{y}{3}, \frac{z}{3} \right) \]

### Step 3: Set the centroid equal to the given point
We know that the centroid G is (1, -2, 3). Therefore, we can set up the following equations:
\[ \frac{x}{3} = 1 \quad (1) \]
\[ \frac{y}{3} = -2 \quad (2) \]
\[ \frac{z}{3} = 3 \quad (3) \]

### Step 4: Solve for x, y, and z
From equation (1):
\[ x = 3 \cdot 1 = 3 \]

From equation (2):
\[ y = 3 \cdot (-2) = -6 \]

From equation (3):
\[ z = 3 \cdot 3 = 9 \]

### Step 5: Write the intercepts
The intercepts of the plane on the axes are:
- x-intercept (A) = 3
- y-intercept (B) = -6
- z-intercept (C) = 9

### Step 6: Write the equation of the plane
The general form of the equation of a plane that intersects the axes at points (a, 0, 0), (0, b, 0), and (0, 0, c) is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
Substituting the values of a, b, and c:
\[ \frac{x}{3} + \frac{y}{-6} + \frac{z}{9} = 1 \]

### Step 7: Clear the denominators
To eliminate the fractions, multiply through by the least common multiple (LCM) of the denominators (which is 18):
\[ 6x - 3y + 2z = 18 \]

### Step 8: Rearranging the equation
Rearranging gives us:
\[ 6x - 3y + 2z - 18 = 0 \]

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