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

Question

Accepted Answer

To determine the locus of the vertex $ C $ of triangle $ \Delta ABC $ given that the centroid $ G $ moves on the line $ 2x + 3y = 1 $, we start by recalling the formula for the centroid of a triangle. The centroid $ G(x, y) $ 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 triangle $ \Delta ABC $ with vertices $ A(2, -3) $, $ B(-2, 1) $, and $ C(x_3, y_3) $, the coordinates of the centroid $ G $ are:

$$
G\left(\frac{2 - 2 + x_3}{3}, \frac{-3 + 1 + y_3}{3}\right) = G\left(\frac{x_3}{3}, \frac{-2 + y_3}{3}\right)
$$

Given that the centroid $ G $ lies on the line $ 2x + 3y = 1 $, we substitute the coordinates of $ G $ into this equation:

$$
2\left(\frac{x_3}{3}\right) + 3\left(\frac{-2 + y_3}{3}\right) = 1
$$

Simplifying this equation:

$$
\frac{2x_3}{3} + \frac{3(-2 + y_3)}{3} = 1
$$

$$
\frac{2x_3}{3} + (-2 + y_3) = 1
$$

$$
\frac{2x_3}{3} - 2 + y_3 = 1
$$

Multiply through by 3 to eliminate the fraction:

$$
2x_3 - 6 + 3y_3 = 3
$$

Rearrange to find the equation of the locus of $ C $:

$$
2x_3 + 3y_3 = 9
$$

Thus, the locus of the vertex $ C $ is the line $ 2x + 3y = 9 $, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2: $ 2x - 3y = 7 $**: This equation does not satisfy the condition derived from the centroid's movement on the line $ 2x + 3y = 1 $. The calculations show that the correct transformation leads to $ 2x + 3y = 9 $.

- **Option 3: $ 3x + 2y = 5 $**: This equation is not consistent with the derived equation for the locus of $ C $. The transformation of the centroid's coordinates does not lead to this equation.

- **Option 4: $ 3x - 2y = 3 $**: Similarly, this equation does not match the derived equation for the locus of $ C $. The calculations clearly show that the correct equation is $ 2x + 3y = 9 $.

Therefore, Option 1 is the correct answer, as it accurately represents the locus of vertex $ C $ based on the given conditions.