What is the equation of a circle that includes a point at (1,-2) and is centered at (4,2)?

Options

A. `(x-2)^2+(y-2)^2=5`
B. `(x-4)^2+(y-5)^2=2`
C. `(x+4)^2+(y-2)^2=5`
D. `(x-4)^2+(y+2)^2=25`

Answer

To find the equation of a circle that includes the point (1, -2) and is centered at (4, 2), we can follow these steps:

### Step 1: Identify the center and the point on the circle
The center of the circle is given as (4, 2), and the point on the circle is (1, -2).

### Step 2: Use the distance formula to find the radius
The radius of the circle can be calculated using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we can let:
- \((x_1, y_1) = (4, 2)\) (the center)
- \((x_2, y_2) = (1, -2)\) (the point on the circle)

Substituting these values into the distance formula:
\[ d = \sqrt{(1 - 4)^2 + (-2 - 2)^2} \]
Calculating the differences:
\[ d = \sqrt{(-3)^2 + (-4)^2} \]
Calculating the squares:
\[ d = \sqrt{9 + 16} \]
Adding the squares:
\[ d = \sqrt{25} \]
Taking the square root:
\[ d = 5 \]
Thus, the radius \(r\) of the circle is 5.

### Step 3: Write the standard equation of the circle
The standard equation of a circle with center \((a, b)\) and radius \(r\) is given by:
\[ (x - a)^2 + (y - b)^2 = r^2 \]
Substituting the values of the center \((4, 2)\) and radius \(5\):
\[ (x - 4)^2 + (y - 2)^2 = 5^2 \]
Calculating \(5^2\):
\[ (x - 4)^2 + (y - 2)^2 = 25 \]

### Final Answer
The equation of the circle is:
\[ (x - 4)^2 + (y - 2)^2 = 25 \]

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