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

Options

A. `20pi` sq. units
B. `25 pi` sq. units
C. `22pi` sq. units
D. `25pi` sq. units

Answer

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

### Step 1: Identify the center and the 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 can use the distance formula between the center of the circle and the point on the circle. The distance formula 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:

\[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} \]

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:

\[ d = 5 \]

So, the radius \(r\) of the circle is 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 the square of the radius:

\[ A = \pi (25) \]

Thus, the area of the circle is:

\[ A = 25\pi \]

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

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