Two points (1,1) and (1,5) lie on the circumference of a circle. Which of the following can be the area of that circle ?

Options

A. `2sqrt2pi`
B. `4pi`
C. `3sqrt3pi`
D. `6.67 pi`

Answer

To find the area of the circle that has the points (1,1) and (1,5) on its circumference, we can follow these steps:

### Step 1: Identify the points
We have two points:
- Point A (1, 1)
- Point B (1, 5)

### Step 2: Calculate the distance between the points
To find the diameter of the circle, we need to calculate the distance between points A and B using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ d = \sqrt{(1 - 1)^2 + (5 - 1)^2} = \sqrt{0 + 4^2} = \sqrt{16} = 4 \]
Thus, the distance AB (which is the diameter D) is 4.

### Step 3: Calculate the radius
The radius \( r \) of the circle is half of the diameter:
\[ r = \frac{D}{2} = \frac{4}{2} = 2 \]

### Step 4: Calculate the area of the circle
The area \( A \) of the circle can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (2^2) = \pi \cdot 4 = 4\pi \]

### Conclusion
The area of the circle can be \( 4\pi \).