The equation `(x+4)^(2)+(y-7)^(2)=25` represents a circle in the xy-plane . Points A and B on the circle are the endpoints of diameter, and point A has coordinates (-4, 2). What are the coordinates of point B?

Options

A. `(-9, 7)`
B. `(-4, 12)`
C. `(-4, 7)`
D. `(1, 7)`

Answer

To find the coordinates of point B, we will follow these steps:

### Step 1: Identify the center and radius of the circle
The equation of the circle is given as:
\[ (x + 4)^2 + (y - 7)^2 = 25 \]
This can be compared to the standard form of a circle's equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

From the given equation, we can see:
- \(h = -4\)
- \(k = 7\)
- \(r^2 = 25\) which gives \(r = 5\)

### Step 2: Find the coordinates of point A
We are given that point A has coordinates:
\[ A(-4, 2) \]

### Step 3: Determine the coordinates of point B
Since A and B are endpoints of the diameter of the circle, the center of the circle is the midpoint of segment AB. The coordinates of the center (h, k) can be calculated as:
\[ \left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) = (h, k) \]
Substituting the known values:
\[ \left(\frac{-4 + x_B}{2}, \frac{2 + y_B}{2}\right) = (-4, 7) \]

### Step 4: Set up the equations
From the x-coordinates:
\[ \frac{-4 + x_B}{2} = -4 \]
Multiplying both sides by 2:
\[ -4 + x_B = -8 \]
Solving for \(x_B\):
\[ x_B = -8 + 4 = -4 \]

From the y-coordinates:
\[ \frac{2 + y_B}{2} = 7 \]
Multiplying both sides by 2:
\[ 2 + y_B = 14 \]
Solving for \(y_B\):
\[ y_B = 14 - 2 = 12 \]

### Step 5: Write the coordinates of point B
Thus, the coordinates of point B are:
\[ B(-4, 12) \]

### Summary
The coordinates of point B are \((-4, 12)\).

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