on a plane are represented the points A (-4,1) B (2,1) and C (a,b) are vertices of a triangle ABC with an area equal to 24. Find the coordinates of point C

Solution:

1. Given:
- Vertex A (-4,1)
- Vertex B (2,1)
- Vertex C (a,b)
- Area of \triangle ABC = 24

2. The formula for the area of a triangle with vertices (x_1, y_1), (x_2, y_2), and (x_3, y_3):
\text{Area} = \frac{1}{2} | x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) |

3. Substitute the coordinates of points A and B into the formula:
\text{Area of} \ \triangle ABC = \frac{1}{2} | -4(1 - b) + 2(b - 1) + a(1 - 1) | = 24

4. Simplify the equation:
\frac{1}{2} | -4 + 4b + 2b - 2 | = 24
\frac{1}{2} | 6b - 6 | = 24

5. Solve for b:
| 6b - 6 | = 48
So,
6b - 6 = 48
6b = 54
b = 9

or

6b - 6 = -48
6b = -42
b = -7

6. The coordinates of point C are (a, 9) or (a, -7).

7. To find a, note that when b = 9 or b = -7, point C lies on a line parallel to x-axis at y = 9 or y = -7. Since point A and point B have the same y-value, any value of a would be valid for them given their symmetry on the x-axis. Therefore, point C can be (a, 9) or (a, -7).

Thus, the coordinates of point C are (a, 9) or (a, -7).

The coordinates of point C will be determined by the specific x-value of a, which can vary since the area remains consistent through the y coordinate variation given. Therefore, we depict point C in terms of a:
- If b = 9, this is the valid unique coordinate for a single straight line area.

Therefore, the coordinates of point C are (a, 9) or (a, -7)