In the xy-plane, a circle with center O passes through the point (2, 0) and has a radius of 4. Which of the following could be the equation of circle O?

Options

A. `(x-2)^(2)+(y+4)^(2)=4`
B. `(x-2)^(2)+(y+4)^(2)=16`
C. `(x-4)^(2)+(y+2)^(2)=16`
D. `(x+2)^(2)+(y-2)^(2)=16`

Answer

To find the equation of the circle with center O that passes through the point (2, 0) and has a radius of 4, we can follow these steps:

### Step 1: Understand the standard form of a circle's equation
The standard form of a circle's equation is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step 2: Identify the center and radius
From the problem, we know:
- The radius \(r = 4\)
- The circle passes through the point (2, 0).

However, we need to determine the center \(O\). Since the center is not explicitly given, we can denote it as \((h, k)\).

### Step 3: Substitute the known values into the equation
Using the point (2, 0) which lies on the circle, we can substitute into the equation:
\[ (2 - h)^2 + (0 - k)^2 = 4^2 \]
This simplifies to:
\[ (2 - h)^2 + k^2 = 16 \]

### Step 4: Check the options provided
Now, we need to check which of the given options could represent this equation. We will evaluate each option by substituting the point (2, 0) into the equations.

**Option A:**
\[ (x - 2)^2 + (y + 4)^2 = 4 \]
Substituting (2, 0):
\[ (2 - 2)^2 + (0 + 4)^2 = 4 \implies 0 + 16 = 4 \quad \text{(False)} \]

**Option B:**
\[ (x - 2)^2 + (y + 4)^2 = 16 \]
Substituting (2, 0):
\[ (2 - 2)^2 + (0 + 4)^2 = 16 \implies 0 + 16 = 16 \quad \text{(True)} \]

**Option C:**
\[ (x - 4)^2 + (y + 2)^2 = 16 \]
Substituting (2, 0):
\[ (2 - 4)^2 + (0 + 2)^2 = 16 \implies 4 + 4 = 8 \quad \text{(False)} \]

**Option D:**
\[ (x + 2)^2 + (y - 2)^2 = 16 \]
Substituting (2, 0):
\[ (2 + 2)^2 + (0 - 2)^2 = 16 \implies 16 + 4 = 20 \quad \text{(False)} \]

### Step 5: Conclusion
The only option that satisfies the condition is **Option B**:
\[ (x - 2)^2 + (y + 4)^2 = 16 \]

### Final Answer:
The equation of circle O is given by **Option B**.

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