AB is a diameter of a circle with centre C = (-2, 5). If A = (3, -7). Find
the coordinates of B.

Answer

To find the coordinates of point B given that AB is the diameter of a circle with center C, we can follow these steps:

### Step 1: Understand the relationship between points A, B, and C
Since C is the center of the circle and also the midpoint of the diameter AB, we can use the midpoint formula. The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
In our case, the midpoint C is given as (-2, 5), and point A is given as (3, -7). We need to find the coordinates of point B, which we will denote as (x, y).

### Step 2: Set up the equations using the midpoint formula
Using the midpoint formula, we can set up two equations based on the x-coordinates and y-coordinates:
1. For the x-coordinates:
\[ C_x = \frac{A_x + B_x}{2} \]
Substituting the known values:
\[ -2 = \frac{3 + x}{2} \]

2. For the y-coordinates:
\[ C_y = \frac{A_y + B_y}{2} \]
Substituting the known values:
\[ 5 = \frac{-7 + y}{2} \]

### Step 3: Solve the equations for x and y
**Solving for x:**
Multiply both sides of the first equation by 2:
\[ -4 = 3 + x \]
Now, isolate x:
\[ x = -4 - 3 = -7 \]

**Solving for y:**
Multiply both sides of the second equation by 2:
\[ 10 = -7 + y \]
Now, isolate y:
\[ y = 10 + 7 = 17 \]

### Step 4: Write the coordinates of point B
Now that we have found the values of x and y, the coordinates of point B are:
\[ B = (-7, 17) \]

### Final Answer
The coordinates of point B are \((-7, 17)\).

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