Question

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

Hint:

When finding the coordinates of a point given its midpoint, use the midpoint formula to solve for the unknown coordinates.

Answer 1

Let the coordinates of point A be \( (x, y) \). Since AB is the diameter of the circle, the center of the circle is the midpoint of AB. The midpoint \( M \) of a line segment joining the points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right). \] Given that the center of the circle is \( (2, -3) \), the midpoint of AB is \( (2, -3) \). The coordinates of point B are \( (1, 4) \), so we can use the midpoint formula to find the coordinates of point A. Let the coordinates of A be \( (x, y) \). The midpoint is: \[ \left( \frac{x + 1}{2}, \frac{y + 4}{2} \right) = (2, -3). \] Equating the coordinates: \[ \frac{x + 1}{2} = 2 \quad \text{and} \quad \frac{y + 4}{2} = -3. \]
Step 1: Solve for \( x \) and \( y \): \[ x + 1 = 4 \quad \Rightarrow \quad x = 3. \] \[ y + 4 = -6 \quad \Rightarrow \quad y = -10. \] Thus, the coordinates of point A are \( (3, -10) \).

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