To solve the problem, we are asked to find the coordinates of point A, where AB is the diameter of a circle. The center of the circle is given as $C(2, -3)$, and point B is $(1, 4)$.

Since the center of a circle lies at the midpoint of the diameter, we can use the midpoint formula to find the coordinates of point A.

The midpoint formula is: $\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$ Where:

• $(x_1, y_1)$ are the coordinates of point A
• $(x_2, y_2)$ are the coordinates of point B
• The midpoint is the center of the circle $C(2, -3)$.

We can set up equations for the x and y coordinates: $\frac{x_A + 1}{2} = 2$ $\frac{y_A + 4}{2} = -3$

Solving for $x_A$:

$\frac{x_A + 1}{2} = 2$ Multiplying both sides by 2: $x_A + 1 = 4$ $x_A = 4 - 1 = 3$

Solving for $y_A$:

$\frac{y_A + 4}{2} = -3$ Multiplying both sides by 2: $y_A + 4 = -6$ $y_A = -6 - 4 = -10$

Thus, the coordinates of point A are $(3, -10)$.

If you'd like any additional details or have further questions, feel free to ask!

Here are 5 related questions:

1. How do you find the length of the diameter of the circle given points A and B?
2. Can you find the equation of the circle if you know the radius and center?
3. How would the midpoint formula apply if AB was not the diameter?
4. What is the general equation for a circle with center $C(h, k)$?
5. How would the process change if you were only given the radius of the circle instead of the diameter?

Tip: The midpoint formula is a useful tool in coordinate geometry, especially when dealing with line segments like diameters and medians.