Question
Note: Your teacher will grade your response to this question to ensure you receive proper credit for your answer. 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.
Answers
GPT 3.5
To find the center of the circle, we first need to find the midpoint of diameter PQ. This can be done by averaging the x-coordinates of P and Q to find the x-coordinate of the center, and averaging the y-coordinates to find the y-coordinate of the center.
Midpoint:
x-coordinate of center = (-10 + 4) / 2 = (-6) / 2 = -3
y-coordinate of center = (-2 + 6) / 2 = 4 / 2 = 2
Therefore, the center of the circle is (-3, 2).
To find the radius, we use the distance formula with one of the endpoints of the diameter and the center we found.
Distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
r = sqrt((-3 - (-10))^2 + (2 - (-2))^2)
r = sqrt(7^2 + 4^2)
r = sqrt(49 + 16)
r = sqrt(65)
The radius of the circle is √65.
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values we found:
(x + 3)^2 + (y - 2)^2 = 65
Therefore, the equation of the circle is (x + 3)^2 + (y - 2)^2 = 65.
Midpoint:
x-coordinate of center = (-10 + 4) / 2 = (-6) / 2 = -3
y-coordinate of center = (-2 + 6) / 2 = 4 / 2 = 2
Therefore, the center of the circle is (-3, 2).
To find the radius, we use the distance formula with one of the endpoints of the diameter and the center we found.
Distance formula:
r = sqrt((x2 - x1)^2 + (y2 - y1)^2)
r = sqrt((-3 - (-10))^2 + (2 - (-2))^2)
r = sqrt(7^2 + 4^2)
r = sqrt(49 + 16)
r = sqrt(65)
The radius of the circle is √65.
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values we found:
(x + 3)^2 + (y - 2)^2 = 65
Therefore, the equation of the circle is (x + 3)^2 + (y - 2)^2 = 65.