The diameter of a circle has endpoints P(- 10, - 8) ) and Q(4, 4) . a. Find the center of the circle. bFind the radius. If your answer is not an integer, express it in radical form. . Write an equation for the circle.
If you are given two points (a,b) and (c,d)
the midpoint is ( (a+c)/2 , (b+d)/2 )
and the distance between them is √( (a-c)^2 + (b-d)^2 )
apply this to your data,
then let me know what you got for the equation of the circle.
the center is the midpoint of PQ
the radius is half the length of PQ
with (h,k) as the center, and r as the radius
... the circle equation is ... (x - h)^2 + (y - k)^2 = r^2
a. To find the center of the circle, we need to find the midpoint of the diameter. The midpoint formula is given by:
Midpoint (x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Using the coordinates of the endpoints P(-10, -8) and Q(4, 4), we can calculate the midpoint as follows:
Midpoint (x, y) = ((-10 + 4)/2, (-8 + 4)/2)
= (-6/2, -4/2)
= (-3, -2)
Therefore, the center of the circle is C(-3, -2).
b. To find the radius of the circle, we need to find the distance between the center and one of the endpoints of the diameter. The distance formula is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of C(-3, -2) and either P(-10, -8) or Q(4, 4), we can calculate the distance as follows:
Distance = √((-3 - (-10))^2 + (-2 - (-8))^2)
= √((7)^2 + (6)^2)
= √(49 + 36)
= √(85)
Therefore, the radius of the circle is √85.
c. The equation for a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Using the coordinates of the center C(-3, -2) and the radius √85, we can write the equation as:
(x - (-3))^2 + (y - (-2))^2 = (√85)^2
(x + 3)^2 + (y + 2)^2 = 85
To find the center of the circle, you can use the midpoint formula. The midpoint of a line segment is the point that is equidistant from both endpoints.
a. Finding the center:
The midpoint formula is given by:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)
Using the endpoints P(-10, -8) and Q(4, 4), we can plug in the values into the formula:
x-coordinate of the midpoint = (-10 + 4) / 2 = -6 / 2 = -3
y-coordinate of the midpoint = (-8 + 4) / 2 = -4 / 2 = -2
Therefore, the center of the circle is C(-3, -2).
b. Finding the radius:
The radius is the distance from the center of the circle to any point on the circle. In this case, we can use the distance formula. The distance formula is given by:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the center C(-3, -2) and any point on the circle, for example, P(-10, -8), we can calculate the distance:
Distance = √((-3 - (-10))^2 + (-2 - (-8))^2)
= √((7)^2 + (6)^2)
= √(49 + 36)
= √85
Therefore, the radius of the circle is √85.
c. Writing the equation for the circle:
The equation for a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Using the center C(-3, -2) and the radius √85, we can write the equation:
(x - (-3))^2 + (y - (-2))^2 = (√85)^2
(x + 3)^2 + (y + 2)^2 = 85