Circles are described below by the coordinates of their centers, C, and one point on their circumference, A. Determine an equati
on for each circle in center-radius form.
(a) C(5,2) and A(11,10)
(b) C(-2,-5) and A(3,-17)
(c) C(5,-1) and A(-2,-5)
1 answer:
Using the given center and point at the circumference, the equation of the circles are:
- a)
![(x - 5)^2 + (y - 2)^2 = 100](https://tex.z-dn.net/?f=%28x%20-%205%29%5E2%20%2B%20%28y%20-%202%29%5E2%20%3D%20100)
- b)
![(x + 2)^2 + (y + 5)^2 = 169](https://tex.z-dn.net/?f=%28x%20%2B%202%29%5E2%20%2B%20%28y%20%2B%205%29%5E2%20%3D%20169)
- c)
![(x - 5)^2 + (y + 1)^2 = 65](https://tex.z-dn.net/?f=%28x%20-%205%29%5E2%20%2B%20%28y%20%2B%201%29%5E2%20%3D%2065)
<h3>
What is the equation of a circle?</h3>
- The equation of a circle of radius r and center
is given by:
![(x - x_0)^2 + (y - y_0)^2 = r^2](https://tex.z-dn.net/?f=%28x%20-%20x_0%29%5E2%20%2B%20%28y%20-%20y_0%29%5E2%20%3D%20r%5E2)
Item a:
- Center C(5,2), hence
.
The point at the circumference is A(11,10), hence:
![(x - 5)^2 + (y - 2)^2 = r^2](https://tex.z-dn.net/?f=%28x%20-%205%29%5E2%20%2B%20%28y%20-%202%29%5E2%20%3D%20r%5E2)
![(11 - 5)^2 + (10 - 2)^2 = r^2](https://tex.z-dn.net/?f=%2811%20-%205%29%5E2%20%2B%20%2810%20-%202%29%5E2%20%3D%20r%5E2)
![r^2 = 100](https://tex.z-dn.net/?f=r%5E2%20%3D%20100)
Hence:
![(x - 5)^2 + (y - 2)^2 = 100](https://tex.z-dn.net/?f=%28x%20-%205%29%5E2%20%2B%20%28y%20-%202%29%5E2%20%3D%20100)
Item b:
- Center C(-2,-5), hence
.
The point at the circumference is A(3,-17), hence:
![(x + 2)^2 + (y + 5)^2 = r^2](https://tex.z-dn.net/?f=%28x%20%2B%202%29%5E2%20%2B%20%28y%20%2B%205%29%5E2%20%3D%20r%5E2)
![(3 + 2)^2 + (-17 - 5)^2 = r^2](https://tex.z-dn.net/?f=%283%20%2B%202%29%5E2%20%2B%20%28-17%20-%205%29%5E2%20%3D%20r%5E2)
![r^2 = 169](https://tex.z-dn.net/?f=r%5E2%20%3D%20169)
Hence:
![(x + 2)^2 + (y + 5)^2 = 169](https://tex.z-dn.net/?f=%28x%20%2B%202%29%5E2%20%2B%20%28y%20%2B%205%29%5E2%20%3D%20169)
Item c:
- Center C(5,-1), hence
.
The point at the circumference is A(-2,-5), hence:
![(x - 5)^2 + (y + 1)^2 = r^2](https://tex.z-dn.net/?f=%28x%20-%205%29%5E2%20%2B%20%28y%20%2B%201%29%5E2%20%3D%20r%5E2)
![(-2 - 5)^2 + (-5 + 1)^2 = r^2](https://tex.z-dn.net/?f=%28-2%20-%205%29%5E2%20%2B%20%28-5%20%2B%201%29%5E2%20%3D%20r%5E2)
![r^2 = 65](https://tex.z-dn.net/?f=r%5E2%20%3D%2065)
Hence:
![(x - 5)^2 + (y + 1)^2 = 65](https://tex.z-dn.net/?f=%28x%20-%205%29%5E2%20%2B%20%28y%20%2B%201%29%5E2%20%3D%2065)
You can learn more about equation of a circle at brainly.com/question/24307696
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