The vertex C of a triangle ABC moves on the line `L-=3x+4y+5=0`. The co-ordinates of the points A and B are `(2,7)` and `(5,8)` The locus of centroid of `DeltaABC` is a line parallel to:

Options

A. AB
B. BC
C. CA
D. L

Answer

To solve the problem, we need to find the locus of the centroid of triangle ABC as the vertex C moves along the line \( L: 3x + 4y + 5 = 0 \). The coordinates of points A and B are given as \( A(2, 7) \) and \( B(5, 8) \).

### Step-by-Step Solution:

1. **Define the Coordinates of Point C**:
Let the coordinates of point C be \( C(x_1, y_1) \).

2. **Centroid Formula**:
The coordinates of the centroid \( G \) of triangle ABC can be calculated using the formula:
\[ G\left(H, K\right) = \left(\frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3}\right) \]
Substituting the coordinates of points A and B:
\[ H = \frac{2 + 5 + x_1}{3} = \frac{7 + x_1}{3} \]
\[ K = \frac{7 + 8 + y_1}{3} = \frac{15 + y_1}{3} \]

3. **Express \( x_1 \) and \( y_1 \) in terms of \( H \) and \( K \)**:
Rearranging the equations for \( H \) and \( K \):
\[ x_1 = 3H - 7 \]
\[ y_1 = 3K - 15 \]

4. **Substitute into the Line Equation**:
Since point C lies on the line \( L: 3x + 4y + 5 = 0 \), we substitute \( x_1 \) and \( y_1 \) into the line equation:
\[ 3(3H - 7) + 4(3K - 15) + 5 = 0 \]
Simplifying this:
\[ 9H - 21 + 12K - 60 + 5 = 0 \]
\[ 9H + 12K - 76 = 0 \]

5. **Rearranging the Locus Equation**:
Rearranging the equation gives us:
\[ 9H + 12K = 76 \]
This can be rewritten in the standard line form:
\[ 9X + 12Y = 76 \]

6. **Finding the Slope**:
The slope of the line \( 9X + 12Y = 76 \) can be found by rewriting it in slope-intercept form \( Y = mx + b \):
\[ 12Y = -9X + 76 \implies Y = -\frac{9}{12}X + \frac{76}{12} \]
The slope \( m \) is:
\[ m = -\frac{3}{4} \]

7. **Finding the Slope of Line L**:
The line \( L: 3x + 4y + 5 = 0 \) can also be rewritten in slope-intercept form:
\[ 4y = -3x - 5 \implies y = -\frac{3}{4}x - \frac{5}{4} \]
The slope of line \( L \) is:
\[ m_L = -\frac{3}{4} \]

8. **Conclusion**:
Since the slope of the locus of the centroid \( \frac{3}{4} \) is equal to the slope of line \( L \), we conclude that the locus of the centroid is parallel to line \( L \).

### Final Answer:
The locus of the centroid of triangle ABC is parallel to line \( L \).