Which of the following is the equation of a circle with center (2, 0) and a radius with endpoint `(5, sqrt(7))`?

Options

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

Answer

To find the equation of a circle with center (2, 0) and a radius defined by the endpoint (5, √7), we can follow these steps:

### Step 1: Identify the center and the endpoint of the radius
The center of the circle is given as (2, 0) and the endpoint of the radius is (5, √7).

### Step 2: Use the distance formula to find the radius
The radius \( r \) can be calculated using the distance formula, which states that the distance \( d \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In our case, the points are (2, 0) and (5, √7). Thus, we can substitute these values into the formula:

\[ r = \sqrt{(5 - 2)^2 + (\sqrt{7} - 0)^2} \]

### Step 3: Calculate the radius
Now, we compute the values:

\[ r = \sqrt{(3)^2 + (\sqrt{7})^2} \]

Calculating further:

\[ r = \sqrt{9 + 7} = \sqrt{16} = 4 \]

### Step 4: Write the equation of the circle
The general equation of a circle with center \((h, k)\) and radius \(r\) is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substituting \(h = 2\), \(k = 0\), and \(r = 4\):

\[ (x - 2)^2 + (y - 0)^2 = 4^2 \]

This simplifies to:

\[ (x - 2)^2 + y^2 = 16 \]

### Final Answer
Thus, the equation of the circle is:

\[ (x - 2)^2 + y^2 = 16 \]