Question
Write an equation of the circle that has a diameter with endpoints 12,3 and -18/3.
Answers
GPT 3.5
First, we need to find the center of the circle by finding the midpoint of the diameter.
The midpoint of a line with endpoints (x1, y1) and (x2, y2) is given by the formula:
((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the endpoints of the diameter are (12, 3) and (-18, 3). So, the center of the circle is:
((12 + (-18)) / 2, (3 + 3) / 2)
= (-6 / 2, 6 / 2)
= (-3, 3)
Now, we need to find the radius of the circle, which is half the length of the diameter. The diameter is the distance between the two endpoints, so we can use the distance formula to find the length of the diameter:
d = √( (x2 - x1)^2 + (y2 - y1)^2 )
d = √( (-18 - 12)^2 + (3 - 3)^2 )
d = √( (-30)^2 + 0^2 )
d = √(900)
d = 30
Therefore, the radius of the circle is half the length of the diameter, which is 30/2 = 15.
Finally, we can write the equation of the circle with center (-3, 3) and radius 15:
(x + 3)^2 + (y - 3)^2 = 15^2
(x + 3)^2 + (y - 3)^2 = 225
So, the equation of the circle is (x + 3)^2 + (y - 3)^2 = 225.
The midpoint of a line with endpoints (x1, y1) and (x2, y2) is given by the formula:
((x1 + x2) / 2, (y1 + y2) / 2)
In this case, the endpoints of the diameter are (12, 3) and (-18, 3). So, the center of the circle is:
((12 + (-18)) / 2, (3 + 3) / 2)
= (-6 / 2, 6 / 2)
= (-3, 3)
Now, we need to find the radius of the circle, which is half the length of the diameter. The diameter is the distance between the two endpoints, so we can use the distance formula to find the length of the diameter:
d = √( (x2 - x1)^2 + (y2 - y1)^2 )
d = √( (-18 - 12)^2 + (3 - 3)^2 )
d = √( (-30)^2 + 0^2 )
d = √(900)
d = 30
Therefore, the radius of the circle is half the length of the diameter, which is 30/2 = 15.
Finally, we can write the equation of the circle with center (-3, 3) and radius 15:
(x + 3)^2 + (y - 3)^2 = 15^2
(x + 3)^2 + (y - 3)^2 = 225
So, the equation of the circle is (x + 3)^2 + (y - 3)^2 = 225.