A circle passes through the points (-1,3) and (5,11) and its radius is 5. Then, its centre is

Options

A. (-5,0)
B. (-5,7)
C. (2,7)
D. (5,0)

Answer

To find the center of the circle that passes through the points (-1, 3) and (5, 11) with a radius of 5, we can follow these steps:

### Step 1: Calculate the distance between the two points
We will use the distance formula to find the distance between the points (-1, 3) and (5, 11).

The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Substituting the coordinates:
\[ d = \sqrt{(5 - (-1))^2 + (11 - 3)^2} \]
\[ d = \sqrt{(5 + 1)^2 + (11 - 3)^2} \]
\[ d = \sqrt{(6)^2 + (8)^2} \]
\[ d = \sqrt{36 + 64} \]
\[ d = \sqrt{100} \]
\[ d = 10 \]

### Step 2: Understand the relationship between the radius and the distance
Since the distance between the two points is 10 and the radius of the circle is 5, this means that the two points are at the ends of a diameter of the circle. Therefore, the center of the circle is the midpoint of the line segment connecting these two points.

### Step 3: Calculate the midpoint
The midpoint \(M\) of a line segment connecting the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting our points:
\[ M = \left( \frac{-1 + 5}{2}, \frac{3 + 11}{2} \right) \]
\[ M = \left( \frac{4}{2}, \frac{14}{2} \right) \]
\[ M = (2, 7) \]

### Conclusion
Thus, the center of the circle is at the point \((2, 7)\).