Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2,-3) and B is (1,4).

Answer

To find the coordinates of point A where AB is the diameter of a circle with center C(2, -3) and point B(1, 4), we can use the midpoint formula. The midpoint of a line segment is the average of the coordinates of its endpoints.

### Step-by-Step Solution:

1. **Understand the Midpoint Formula**:
The midpoint \( C \) of a line segment with endpoints \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:
\[ C = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

2. **Assign Known Values**:
We know:
- The coordinates of point B are \( B(1, 4) \).
- The coordinates of the center (which is also the midpoint) are \( C(2, -3) \).
- Let the coordinates of point A be \( A(x, y) \).

3. **Set Up the Equations**:
Using the midpoint formula for the x-coordinates:
\[ 2 = \frac{x + 1}{2} \]
For the y-coordinates:
\[ -3 = \frac{y + 4}{2} \]

4. **Solve for x**:
Multiply both sides of the first equation by 2:
\[ 4 = x + 1 \]
Subtract 1 from both sides:
\[ x = 3 \]

5. **Solve for y**:
Multiply both sides of the second equation by 2:
\[ -6 = y + 4 \]
Subtract 4 from both sides:
\[ y = -10 \]

6. **Final Coordinates of Point A**:
Thus, the coordinates of point A are:
\[ A(3, -10) \]

### Summary:
The coordinates of point A are \( (3, -10) \).