A diameter of a circle has endpoints P(–10, –2) and Q(4, 6).
Find the center of the circle.
Find the radius. If your answer is not an integer, express it in radical form.
Write an equation for the circle.
To find the center of the circle, we can use the midpoint formula, which states that the center is the midpoint of the line segment formed by the endpoints of the diameter.
Center coordinates = ((-10 + 4)/2, (-2 + 6)/2)
= (-6/2, 4/2)
= (-3, 2)
So, the center of the circle is at (-3, 2).
Next, we can find the radius of the circle by finding the distance between one of the endpoints of the diameter and the center. We can use the distance formula for this. Let's use point P (-10, -2) to find the radius.
r = √[(-10 - (-3))^2 + (-2 - 2)^2]
= √[(-10 + 3)^2 + (-4)^2]
= √[(-7)^2 + 16]
= √[49 + 16]
= √65
So, the radius of the circle is √65.
The equation of a circle with center (h, k) and radius r is given by:
(x-h)^2 + (y-k)^2 = r^2
Substitute the center (-3, 2) and radius √65 into the formula:
(x+3)^2 + (y-2)^2 = (√65)^2
(x+3)^2 + (y-2)^2 = 65
Therefore, the equation of the circle is (x+3)^2 + (y-2)^2 = 65.