The area of the circle centred at (1,2) and passing through (4,6) is

Options

A. `5pi`
B. `10pi`
C. `25pi`
D. none of these

Explanation

To find the area of the circle centered at (1, 2) and passing through (4, 6), we can follow these steps:

### Step 1: Identify the center and a point on the circle
The center of the circle is given as \( (1, 2) \) and a point on the circle is \( (4, 6) \).

### Step 2: Calculate the radius of the circle
To find the radius, we will use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( (x_1, y_1) = (1, 2) \) and \( (x_2, y_2) = (4, 6) \).

Substituting the values into the distance formula:
\[ r = \sqrt{(4 - 1)^2 + (6 - 2)^2} \]
Calculating the differences:
\[ r = \sqrt{(3)^2 + (4)^2} \]
Calculating the squares:
\[ r = \sqrt{9 + 16} \]
Adding the results:
\[ r = \sqrt{25} \]
Taking the square root:
\[ r = 5 \]

### Step 3: Calculate the area of the circle
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
Substituting the radius we found:
\[ A = \pi (5)^2 \]
Calculating \( 5^2 \):
\[ A = \pi \cdot 25 \]
Thus, the area of the circle is:
\[ A = 25\pi \]

### Final Answer
The area of the circle is \( 25\pi \).

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