GEOMETRY
EQUATION OF A CIRCLE
Sept. 13, 2011
We showed in class that the distance between two points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) is given by the Distance
Formula:
𝐷 = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
4
You may also recall the definition of a circle: the set of all points that are the same
(fixed) distance from a given point (the center of the circle). The fixed distance is,
of course, the radius of the circle, which we will call 𝑟. Let’s look at the simple
example of a circle with its center at the origin (0,0). For any point on the circle,
with coordinates (𝑥, 𝑦), we can apply the distance formula, substituting the radius
(x,y)
r
2
(0,0)
-5
5
-2
𝑟 for the distance: 𝑟 = √(𝑥 − 0)2 + (𝑦 − 0)2 . If we square both sides, to
-4
eliminate the radical, we’re left with a nice simple equation:
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𝑥 2 + 𝑦 2 = 𝑟 2.
Circle with radius 𝑟, center at the origin:
(x,y)
6
Now, let’s consider the case where the center is not at the origin, but at a point
with coordinates (𝑎, 𝑏). Again, we use the distance formula to show the fixed
distance from the point (𝑥, 𝑦) to the center (𝑎, 𝑏) to be equal to the radius, 𝑟:
𝑟 = √(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 .
general equation of a circle:
(0,0)
(𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑟 2 .
(why is there a + sign?)
Determine the center coordinates and radius for the circle given by (𝑥 + 4)2 + (𝑦 − 1)2 = 40.
Answer: center (−4, 1); radius = √40 = 2√10
Ex 3:
-2
Write the equation of the circle with radius 5 and center at (2, −7).
Answer: (𝑥 − 2)2 + (𝑦 + 7)2 = 25
Ex 2:
(a,b)
2
Again, if we square both sides, we get the
Circle with radius 𝑟 and center (𝑎, 𝑏):
Ex 1:
r
4
(remember to simplify radicals)
Write the equation of the circle that has 𝐴(1, −6) and 𝐵(5, 2) as endpoints of a diameter.
Method 1:
Determine the center using the Midpoint Formula:
1+5 −6+2
, 2 )
2
𝐶(
→ 𝐶(3, −2)
Determine the radius using the distance formula (center and end of diameter):
𝑟 = √(3 − 1)2 + (−2 + 6)2 = √4 + 16 = √20 = 2√5
Equation of circle is:
(𝑥 − 3)2 + (𝑦 + 2)2 = 20
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Method 2:
Determine center using Midpoint Formula (as before): 𝐶(3, −2).
Thus, the circle equation will have the form (𝑥 − 3)2 + (𝑦 + 2)2 = 𝑟 2
Find 𝑟 2 by plugging the coordinates of a point on the circle in for 𝑥 𝑎𝑛𝑑 𝑦.
Let’s use 𝐵(5, 2): 𝑟 2 = (5 − 3)2 + (2 + 2)2 = 22 + 42 = 4 + 16 = 20
Again, we get this equation for the circle: (𝑥 − 3)2 + (𝑦 + 2)2 = 20
Homework Problems:
A.
B.
Determine the center and radius for the circles given by the following equations. Then sketch the
graph of the circle.
1.
𝑥 2 + 𝑦 2 = 49
4.
(𝑥 + 4)2 + (𝑦 − 2)2 = 52
5.
(𝑥 − 2)2 + (𝑦 + 6)2 =
4
6.
(𝑥 + 3)2 + (𝑦 + 9)2 = 200
7.
(𝑥 − 4)2 + (𝑦 + 4)2 = 16
3.
(𝑥 + 8)2 + (𝑦 + 1)2 = 72
1
Write an equation for each of the circles described below.
8.
C.
2. 𝑥 2 + (𝑦 − 3)2 = 1
Center (2, 0); radius = 2
9. Center (6, −5); radius = √41
10. Center (−7, 2); radius = 3√5
11. Center (−1, −6); radius = 4√2
12. Center (8, 4); diameter = 5
13. Center (3, 3); diameter = 6√3
14. Center (−3, 7); point on circle (0, 5)
15. Center (3, −2); point on circle (6, 2)
16. Endpoints of diameter: (1, 5) and (−7, −5)
17. Endpoints of diameter: (−4, 2) and (6, 8)
18. Endpoints of diameter: (0, 12) and (−6, 0)
19. Endpoints of diameter: (5, −1) and (3, 3)
Misc. Problems: make a sketch for each problem.
20. Show that the triangle with vertices 𝐴(−3, 4), 𝑀(3, 1), and 𝑌(0, −2) is isosceles.
21. Find the area of the rectangle with vertices 𝐵(8, 0), 𝑇(2, −9), 𝑅(−1, −7), and 𝐶(5, 2).
22. Sketch the graphs of:
a. (𝑥 − 4)2 + (𝑦 + 4)2 ≤ 16
b. (𝑥 + 2)2 + (𝑦 − 3)2 > 1
D. Simplify each of the following radicals by factoring out all perfect squares:
23. √1000
24. √1296
25. √864
26. √1512