Let A (2,-3) and B(-2,1) be vertices of a triangle ABC. If the centroid of this triangle moves on line 2x + 3y = 1, then the locus of the vertex C is the line :

Explanation

To find the locus of vertex C of triangle ABC given that the centroid G of the triangle moves along the line \(2x + 3y = 1\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A and B:**
- Let \(A(2, -3)\) and \(B(-2, 1)\) be the vertices of the triangle.

2. **Determine the Coordinates of the Centroid G:**
- The formula for the centroid \(G\) of a triangle with vertices \(A(x_1, y_1)\), \(B(x_2, y_2)\), and \(C(x_3, y_3)\) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
- For our triangle, substituting the coordinates of \(A\) and \(B\) and letting \(C(h, k)\):
\[ G\left(\frac{2 + (-2) + h}{3}, \frac{-3 + 1 + k}{3}\right) = G\left(\frac{h}{3}, \frac{k - 2}{3}\right) \]

3. **Substitute the Centroid Coordinates into the Line Equation:**
- We know that the centroid lies on the line \(2x + 3y = 1\). Substituting the coordinates of \(G\):
\[ 2\left(\frac{h}{3}\right) + 3\left(\frac{k - 2}{3}\right) = 1 \]
- Simplifying this equation:
\[ \frac{2h}{3} + \frac{3(k - 2)}{3} = 1 \]
\[ 2h + 3(k - 2) = 3 \]

4. **Simplify the Equation:**
- Expanding the equation:
\[ 2h + 3k - 6 = 3 \]
- Rearranging gives:
\[ 2h + 3k = 9 \]

5. **Identify the Locus of Vertex C:**
- The equation \(2h + 3k = 9\) represents a line in the coordinate plane, where \(h\) and \(k\) are the coordinates of point \(C\).
- Therefore, the locus of vertex \(C\) is given by the line:
\[ 2x + 3y = 9 \]

### Final Answer:
The locus of vertex \(C\) is the line \(2x + 3y = 9\).