Question

The centre \( (2, -3) \) of a circle has diameter as AB. The coordinate of B is \( (1, 4) \). The coordinate of A will be:

• A. \( (3, 10) \)
• B. \( (10, 3) \)
• C. \( (-10, 3) \)
• D. \( (3, -10) \)

Hint:

To find the other endpoint of a line segment, use the midpoint formula and solve for the unknown coordinates.

Answer 1

The Correct Option is D

Step 1: Midpoint formula.
The centre of the circle is the midpoint of the diameter \( AB \). Given the midpoint \( (2, -3) \) and the coordinates of \( B(1, 4) \), we use the midpoint formula to find the coordinates of \( A \). The midpoint formula is: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \), respectively.
Step 2: Apply the formula.
Let the coordinates of \( A \) be \( (x, y) \). The midpoint formula becomes: \[ \left( \frac{x + 1}{2}, \frac{y + 4}{2} \right) = (2, -3) \]
Step 3: Solve for \( x \) and \( y \).
Equating the x-coordinates and y-coordinates: \[ \frac{x + 1}{2} = 2 \quad \Rightarrow \quad x + 1 = 4 \quad \Rightarrow \quad x = 3 \] \[ \frac{y + 4}{2} = -3 \quad \Rightarrow \quad y + 4 = -6 \quad \Rightarrow \quad y = -10 \]
Step 4: Conclusion.
The coordinates of \( A \) are \( (3, -10) \). Therefore, the correct answer is (D).