This one is confusing me. please help
The 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.
Thanks
Surely you can see that the center of the circle is the midpoint of any diameter.
the midpoint R = (P+Q)/2 = (-3,2)
the radius is half the diameter
d^2 = 14^2 + 8^2 = 196+64 = 260
r^2 = 65
(x+3)^2 + (y-2)^2 = 65
To find the center of the circle, we can use the midpoint formula. The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) can be found by taking the average of the x-coordinates and the average of the y-coordinates.
In this case, the x-coordinates of the endpoints are -10 and 4, and the y-coordinates are -2 and 6. Taking the averages, we get:
x-coordinate of the center = (-10 + 4) / 2 = -6/2 = -3
y-coordinate of the center = (-2 + 6) / 2 = 4/2 = 2
So the center of the circle is C(-3, 2).
To find the radius of the circle, we can use the distance formula. The distance between two points (x1, y1) and (x2, y2) is given by the formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the coordinates of the center C(-3, 2) and endpoint P(-10, -2) can be used to find the distance:
distance = sqrt((-10 - (-3))^2 + (-2 - 2)^2)
= sqrt((-7)^2 + (-4)^2)
= sqrt(49 + 16)
= sqrt(65)
So the radius of the circle is sqrt(65).
Finally, to write the equation of the circle, we can use the standard form of the equation:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the coordinates of the center and r represents the radius.
Plugging in the values we found, the equation of the circle is:
(x + 3)^2 + (y - 2)^2 = (sqrt(65))^2
(x + 3)^2 + (y - 2)^2 = 65