A plane meets the coordinates axes in A, B, C such that the centroid of the triangle ABC is the point (a,a,a) .Then the equation of the plane is `x+y+z=p`, where p is ___

Options

A. a
B. `3//a`
C. `a//3`
D. 3a

Answer

To solve the problem, we need to find the value of \( p \) in the equation of the plane \( x + y + z = p \) given that the centroid of the triangle formed by the intercepts of the plane on the coordinate axes is the point \( (a, a, a) \).

### Step-by-step Solution:

1. **Identify the Points of Intersection**:
- The plane intersects the x-axis at point \( A \) with coordinates \( (p, 0, 0) \).
- The plane intersects the y-axis at point \( B \) with coordinates \( (0, q, 0) \).
- The plane intersects the z-axis at point \( C \) with coordinates \( (0, 0, r) \).

2. **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 \( C \):
\[ G = \left( \frac{p + 0 + 0}{3}, \frac{0 + q + 0}{3}, \frac{0 + 0 + r}{3} \right) = \left( \frac{p}{3}, \frac{q}{3}, \frac{r}{3} \right) \]

3. **Setting the Centroid Equal to Given Point**:
- We know that the centroid is \( (a, a, a) \). Therefore, we can set up the following equations:
\[ \frac{p}{3} = a, \quad \frac{q}{3} = a, \quad \frac{r}{3} = a \]

4. **Solving for p, q, and r**:
- From the equations above, we can express \( p \), \( q \), and \( r \) in terms of \( a \):
\[ p = 3a, \quad q = 3a, \quad r = 3a \]

5. **Equation of the Plane**:
- The equation of the plane in intercept form is given by:
\[ \frac{x}{p} + \frac{y}{q} + \frac{z}{r} = 1 \]
- Substituting \( p \), \( q \), and \( r \):
\[ \frac{x}{3a} + \frac{y}{3a} + \frac{z}{3a} = 1 \]
- Multiplying through by \( 3a \):
\[ x + y + z = 3a \]

6. **Conclusion**:
- Comparing this with the given equation \( x + y + z = p \), we find that:
\[ p = 3a \]

### Final Answer:
Thus, the value of \( p \) is \( 3a \).