Question

The points (0, -8) and (10, 2) represent the endpoints of a diameter of a circel. Which of the following reprezents the equation of this circle

Answer 1

The center of the circle can be found by finding the midpoint of the diameter formed by the given endpoints (0, -8) and (10, 2). The midpoint is the average of the x-coordinates and the y-coordinates.

Midpoint: \((\frac{0 + 10}{2}, \frac{-8 + 2}{2}) = (5, -3)\)

Now, the radius of the circle is the distance from the center to one of the endpoints. Using the distance formula:

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

Let's use (0, -8) as one endpoint:

\[ r = \sqrt{(0 - 5)^2 + (-8 - (-3))^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} \]

Now, the equation of a circle with center \((h, k)\) and radius \(r\) is:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substitute the values:

\[ (x - 5)^2 + (y + 3)^2 = 50 \]

So, the equation of the circle is \( (x - 5)^2 + (y + 3)^2 = 50 \).