To find the center of the circle, we first need to find the midpoint of the line segment PQ, as the center of the circle will be equidistant from both endpoints of the diameter. The midpoint M can be calculated using the midpoint formula:
M(x, y) = ((x_1 + x_2) / 2, (y_1 + y_2) / 2)
Substitute the coordinates of P and Q into the formula:
M(x, y) = ((-10 + 4) / 2, (-2 + 6) / 2)
M(x, y) = (-6 / 2, 4 / 2)
M(x, y) = (-3, 2)
Therefore, the center of the circle is at C(-3, 2).
Next, we need to calculate the radius of the circle. The radius is equal to half the length of the diameter, which can be found using the distance formula between points P and C:
r = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)
Substitute the coordinates of P and C into the formula:
r = sqrt((-3 - (-10))^2 + (2 - (-2))^2)
r = sqrt(7^2 + 4^2)
r = sqrt(49 + 16)
r = sqrt(65)
Therefore, the radius of the circle is sqrt(65).
Finally, we can write the equation of the circle in standard form. The standard form of the equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Substitute the center (h, k) = (-3, 2) and radius r = sqrt(65) into the equation:
(x + 3)^2 + (y - 2)^2 = 65
Therefore, the equation of the circle is (x + 3)^2 + (y - 2)^2 = 65.
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. (no plagiarism)
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