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 need to find the midpoint of the diameter. We can use the midpoint formula:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Plugging in the coordinates of P and Q:
Midpoint = ((-10 + 4)/2, (-2 + 6)/2) = (-3, 2)
So the center of the circle is (-3, 2).
To find the radius, we can use the distance formula between the center and one of the endpoints of the diameter (e.g. P):
radius = distance(P, center)
= √((x2 – x1)^2 + (y2 – y1)^2)
= √((-3 – (-10))^2 + (2 – (-2))^2)
= √(49 + 16)
= √65
So the radius is √65.
To write the equation of the circle, we can use the standard form:
(x – h)^2 + (y – k)^2 = r^2
where (h, k) is the center of the circle and r is the radius. Plugging in the values we found:
(x – (-3))^2 + (y – 2)^2 = (√65)^2
Simplifying:
(x + 3)^2 + (y – 2)^2 = 65
To find the center of the circle, we can use the midpoint formula.
1) Find the midpoint:
The formula for the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Using this formula, we can find the midpoint of PQ:
Midpoint = ((-10 + 4) / 2, (-2 + 6) / 2)
Midpoint = (-6 / 2, 4 / 2)
Midpoint = (-3, 2)
Therefore, the center of the circle is (-3, 2).
2) Find the radius:
The formula for the distance between two points (x1, y1) and (x2, y2) is:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we can find the distance between P and Q, which is equal to the diameter of the circle:
Distance = √((4 - (-10))^2 + (6 - (-2))^2)
Distance = √((4 + 10)^2 + (6 + 2)^2)
Distance = √(14^2 + 8^2)
Distance = √(196 + 64)
Distance = √260
Therefore, the radius of the circle is √260.
3) Write an equation for the circle:
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Substituting the center (-3, 2) and the radius √260 into the equation, we get:
(x - (-3))^2 + (y - 2)^2 = (√260)^2
(x + 3)^2 + (y - 2)^2 = 260