Key Concepts and Formulas

• Centroid of a Triangle: The centroid of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is given by $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$.
• Locus: The set of all points satisfying a given condition. To find the locus, express the condition in terms of the coordinates of a general point $(x, y)$ and simplify the resulting equation.

Step-by-Step Solution

1. Define the vertices and the centroid. We are given the vertices $A(2, -3)$ and $B(-2, 1)$. Let the third vertex be $C(h, k)$. We aim to find the locus of this vertex. Let the centroid of triangle $ABC$ be $G(x_G, y_G)$. Why? This step sets up the problem by defining the given information and the unknown we want to find. Using $(h, k)$ for vertex $C$ allows us to find a general relationship and ultimately the locus.

2. Calculate the coordinates of the centroid in terms of $(h, k)$. Using the centroid formula, we have: $x_G = \frac{2 + (-2) + h}{3} = \frac{h}{3}$ $y_G = \frac{-3 + 1 + k}{3} = \frac{k - 2}{3}$ So, the centroid is $G\left(\frac{h}{3}, \frac{k-2}{3}\right)$. Why? This step expresses the centroid's coordinates in terms of the unknown coordinates of vertex $C$. This is crucial for using the given condition about the centroid.

3. Apply the condition that the centroid lies on the line $2x + 3y = 1$. Since the centroid $G\left(\frac{h}{3}, \frac{k-2}{3}\right)$ lies on the line $2x + 3y = 1$, we can substitute the coordinates of the centroid into the equation of the line: $2\left(\frac{h}{3}\right) + 3\left(\frac{k-2}{3}\right) = 1$ Why? This step uses the given information to create an equation involving $h$ and $k$, which will help us determine the relationship between them.

4. Simplify the equation to find the locus of vertex $C$. Simplifying the equation, we get: $\frac{2h}{3} + \frac{3(k-2)}{3} = 1$ $\frac{2h}{3} + k - 2 = 1$ Multiplying both sides by 3 to eliminate the fraction: $2h + 3(k - 2) = 3$ $2h + 3k - 6 = 3$ $2h + 3k = 9$ Replacing $h$ with $x$ and $k$ with $y$ to obtain the locus, we get: $2x + 3y = 9$ Why? This step simplifies the equation to a standard form representing the locus of point $C$. Replacing $(h,k)$ with $(x,y)$ gives the general equation of the locus.

Common Mistakes & Tips

• Incorrect Centroid Formula: Ensure you remember the correct formula for the centroid. A common mistake is to add the coordinates and divide by 2 instead of 3.
• Not Replacing (h, k) with (x, y): Remember to replace the temporary variables $(h, k)$ with $(x, y)$ at the end to express the locus in standard form.
• Algebra Errors: Be careful with algebraic manipulations, especially when dealing with fractions.

Summary

We found the locus of vertex $C$ by first expressing the coordinates of the centroid in terms of the coordinates of $C$. Then, we used the given condition that the centroid lies on the line $2x + 3y = 1$ to form an equation. Finally, we simplified the equation and replaced the temporary variables with $x$ and $y$ to obtain the equation of the locus: $2x + 3y = 9$.

Final Answer

The final answer is \boxed{2x + 3y = 9}, which corresponds to option (D).