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

Options

A. (2, 3)
B. (3, -10)
C. (1, -4)
D. (0, -6)

Answer

To find the coordinates of point A where AB is the diameter of a circle with center C at (2, -3) and point B at (1, 4), we can follow these steps:

### Step 1: Understand the relationship between A, B, and C
Since AB is the diameter of the circle, point C (the center of the circle) is the midpoint of segment AB. Therefore, we can use the midpoint formula to express this relationship.

### Step 2: Use the midpoint formula
The midpoint M of a segment with endpoints A(x1, y1) and B(x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
In our case, M is the center C(2, -3) and B is (1, 4). Let A be (x, y).

### Step 3: Set up the equations
From the midpoint formula, we have:
\[ C = \left( \frac{x + 1}{2}, \frac{y + 4}{2} \right) \]
Setting this equal to the coordinates of C:
\[ \left( \frac{x + 1}{2}, \frac{y + 4}{2} \right) = (2, -3) \]

### Step 4: Solve for x and y
This gives us two equations:
1. \( \frac{x + 1}{2} = 2 \)
2. \( \frac{y + 4}{2} = -3 \)

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

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

### Step 5: Write the coordinates of A
Thus, the coordinates of point A are:
\[ A(3, -10) \]

### Final Answer:
The coordinates of A are (3, -10).

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