The area of the circle passing through (-2,6) and having its centre at (1,2) is

Options

A. ` 15 pi ,` sq.units
B. `12 pi , sq` units
C. ` 14pi sq.` units
D. ` 25 pi , ` sq. units

Answer

To find the area of the circle that passes through the point (-2, 6) and has its center at (1, 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 (1, 2).
- The point through which the circle passes is (-2, 6).

### Step 2: Use the distance formula to find the radius
The radius \( r \) of the circle can be determined by calculating the distance between the center of the circle and the point on the circle using the distance formula:

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

Where \( (x_1, y_1) \) is the center (1, 2) and \( (x_2, y_2) \) is the point (-2, 6).

### Step 3: Substitute the coordinates into the distance formula
Substituting the coordinates into the formula:

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

### Step 4: Simplify the expression
Calculating the differences:

\[ r = \sqrt{(-3)^2 + (4)^2} \]

Calculating the squares:

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

Adding the values:

\[ r = \sqrt{25} \]

### Step 5: Calculate the radius
Taking the square root:

\[ r = 5 \]

### Step 6: Use the radius to find 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:

\[ A = \pi \times 25 \]

### Step 7: Finalize the area
Thus, the area of the circle is:

\[ A = 25\pi \text{ square units} \]

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

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