Points (–1, 6) and (–3, 2) are endpoints of the diameter of a circle.
(a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________
(b) What is the center of the circle? Answer: ____________
(c) What is the equation of the circle? Answer: ___________________________
To find the exact length of the diameter, you can use the distance formula. The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the diameter.
(a) Let's calculate the length of the diameter using the distance formula:
Given points:
P1 = (-1, 6)
P2 = (-3, 2)
Using the distance formula:
d = √((-3 - (-1))^2 + (2 - 6)^2)
= √((-3 + 1)^2 + (2 - 6)^2)
= √((-2)^2 + (-4)^2)
= √(4 + 16)
= √20
The exact length of the diameter is √20.
(b) To find the center of the circle, we can simply find the midpoint of the line segment connecting the two endpoints of the diameter. The midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Using the given points:
P1 = (-1, 6)
P2 = (-3, 2)
Midpoint = ((-1 + (-3))/2, (6 + 2)/2)
= ((-4)/2, 8/2)
= (-2, 4)
Therefore, the center of the circle is (-2, 4).
(c) To find the equation of the circle, we can use the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the coordinates of the center of the circle, and r represents the radius of the circle. The radius is half the length of the diameter.
Using the center of the circle (-2, 4) and the diameter length (√20), we can find the equation of the circle:
(x - (-2))^2 + (y - 4)^2 = (√20/2)^2
(x + 2)^2 + (y - 4)^2 = 10
Therefore, the equation of the circle is (x + 2)^2 + (y - 4)^2 = 10.