A(2, 5), B(4, -11) are two fixed points and C is a point which moves on the line `3x+4y+5=0`. Find the locus of the centroid of the triangle ABC.

Answer

To find the locus of the centroid of triangle ABC where A(2, 5), B(4, -11), and C is a point on the line \(3x + 4y + 5 = 0\), we will follow these steps:

### Step 1: Identify the coordinates of points A and B
We have:
- Point A: \( A(2, 5) \)
- Point B: \( B(4, -11) \)

Let the coordinates of point C be \( C(x, y) \).

### Step 2: Write the formula for the centroid of triangle ABC
The centroid \( G \) of triangle ABC is given by the formula:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Substituting the coordinates of A, B, and C:
\[ G\left(\frac{2 + 4 + x}{3}, \frac{5 + (-11) + y}{3}\right) = G\left(\frac{6 + x}{3}, \frac{-6 + y}{3}\right) \]

### Step 3: Express the coordinates of the centroid in terms of h and k
Let \( G(h, k) \) be the coordinates of the centroid. Then:
\[ h = \frac{6 + x}{3} \quad \text{and} \quad k = \frac{-6 + y}{3} \]

### Step 4: Solve for x and y in terms of h and k
From the equations:
1. \( 3h = 6 + x \) implies \( x = 3h - 6 \)
2. \( 3k = -6 + y \) implies \( y = 3k + 6 \)

### Step 5: Substitute x and y into the line equation
We know that point C lies on the line \( 3x + 4y + 5 = 0 \). Substituting \( x \) and \( y \):
\[ 3(3h - 6) + 4(3k + 6) + 5 = 0 \]
Expanding this gives:
\[ 9h - 18 + 12k + 24 + 5 = 0 \]
Combining like terms:
\[ 9h + 12k + 11 = 0 \]

### Step 6: Rearranging the equation
Rearranging gives us the locus of the centroid:
\[ 9h + 12k + 11 = 0 \]
This can be rewritten as:
\[ 9x + 12y + 11 = 0 \]

### Final Answer
The locus of the centroid of triangle ABC is given by the equation:
\[ 9x + 12y + 11 = 0 \]