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

Answer

To find the coordinates of point A, where AB is the diameter of the circle with center C (2, -3) and point B is (3, 4), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The center of the circle (C) is the midpoint of the diameter AB. Therefore, we can use the midpoint formula to find the coordinates of point A.

2. **Midpoint Formula**:
The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
Here, C (the center) is the midpoint, so we can set:
\[ C(2, -3) = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right) \]

3. **Substituting Known Values**:
We know B is (3, 4). Let A be (x_A, y_A). Therefore:
\[ 2 = \frac{x_A + 3}{2} \quad \text{and} \quad -3 = \frac{y_A + 4}{2} \]

4. **Solving for x_A**:
From the first equation:
\[ 2 = \frac{x_A + 3}{2} \]
Multiply both sides by 2:
\[ 4 = x_A + 3 \]
Subtract 3 from both sides:
\[ x_A = 1 \]

5. **Solving for y_A**:
From the second equation:
\[ -3 = \frac{y_A + 4}{2} \]
Multiply both sides by 2:
\[ -6 = y_A + 4 \]
Subtract 4 from both sides:
\[ y_A = -10 \]

6. **Final Coordinates of Point A**:
Thus, the coordinates of point A are:
\[ A(1, -10) \]

### Conclusion:
The coordinates of point A are (1, -10).