1. Geometry
Equations of a Circle

2. Goals Write the equation of a circle.
Use the equation of a circle to graph the circle on the coordinate plane.
Solve problems with circles.
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3. Circle Definition
A circle is the set of points on a plane that are equidistant from the center.
The radius, r, is the distance between the center (h, k) and any point (x, y) on the circle.
(x, y) r
(h, k)
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4. Circle Equation Use the Distance Formula to write this. (x, y) r
(h, k)
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5. Circle Equation
(x, y)
Square both sides: r
(h, k)
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6. The Equation of a Circle
Where:
(h, k) is the center
r is the radius
(x, y) is any point on the circle
(x, y) r
(h, k)
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7. What is the center and radius?
(x – 9)2 + (y – 1)2 = 25
Center: (9, 1)
Radius: 5
(x – 9)2 + (y – 1)2 = 52
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8. What is the center and radius?
(x – 2)2 + (y + 1)2 = 1
(x – 2)2 + (y – (-1))2 = 12
Center: (2, -1)
Radius: 1
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9. What is the center and radius?
(x – 6)2 + y2 = 100
Center: (6, 0)
Radius: 10
(x – 6)2 + (y – 0)2 = 102
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10. Your Turn Identify the center and radius of each circle:
(x – 12)2 + (y + 3)2 = 4
Center: (12, –3) Radius = 2
x2 + y2 = 121
Center: (0, 0) Radius = 11
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11. Example 42 = (x – 5)2 + (y – 6)2 16 = (x – 5)2 + (y – 6)2
Write the equation of a circle with center (5, 6) and radius 4.
42 = (x – 5)2 + (y – 6)2
16 = (x – 5)2 + (y – 6)2
or (x – 5)2 + (y – 6)2 = 16
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12. Your Turn 82 = (x – 1)2 + (y – (-3))2 (x – 1)2 + (y + 3)2 = 64
Write the equation of a circle with center (1, -3) and radius 8.
82 = (x – 1)2 + (y – (-3))2
(x – 1)2 + (y + 3)2 = 64
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13. What if we don’t know r?
The point (3, 2) is on a circle with center (5, 4). Write the equation.
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14. What if we don’t know r? r2 = (3 – 5)2 + (2 – 4)2 r2 = (–2 )2 + (–2)2
The point (3, 2) is on a circle with center (5, 4). Write the equation.
r2 = (3 – 5)2 + (2 – 4)2
r2 = (–2 )2 + (–2)2
r2 = = 8
DON’T SIMPLIFY!
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15. Write the equation. r2 = 8 (x – 5)2 + (y – 4)2 = 8
The point (3, 2) is on a circle with center (5, 4). Write the equation.
r2 = 8
(x – 5)2 + (y – 4)2 = 8
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16. Your Turn. r2 = (-1 – 2)2 + (4 – 3)2 r2 = (-3)2 + (1)2 r2 = 9 + 1 = 10
The point (-1, 4) is on a circle with center (2, 3). Write the equation.
r2 = (-1 – 2)2 + (4 – 3)2
r2 = (-3)2 + (1)2
r2 = = 10
(x – 2)2 + (y – 3)2 = 10
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17. Graphing Circles Graph the circle given by the equation
(x – 2)2 + (y – 1)2 = 9
First find the center (h, k).
What is h? 2
What is k? 1
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18. Graphing Circles continued
(x – 2)2 + (y – 1)2 = 9
Center (2, 1)
What is r? 3
Why?
(x – 2)2 + (y – 1)2 = 32
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19. Graphing Circles
Knowing the center is (2, 1) and the radius is 3. Graph the circle.
Draw the center.
Draw points at the ends of 4 radii.
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20. Graphing Circles
Knowing the center is (2, 1) and the radius is 3. Graph the circle.
Draw the center.
Draw points at the ends of 4 radii.
Sketch the circle.
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21. Graphing Circles
Knowing the center is (2, 1) and the radius is 3. Graph the circle.
Draw the center.
Draw points at the ends of 4 radii.
Sketch the circle.
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22. Your Turn Graph: (x – 1)2 + (y + 3)2 = 16 Solution: Center: (1, -3)
Radius: 4
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23. Problem
(x + 1)2 + (y – 1)2 = 25
Is the point (3, 4) on the circle, in its interior, or in the exterior?
Directions: Make a sketch of the circle. Then locate (3, 4) and answer the question.
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24. Graphical Solution On the circle. Graph: (x + 1)2 + (y – 1)2 = 25
Center: (-1, 1)
Radius: 5
Locate (3, 4)
On the circle.
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25. What about (3, 2)?
In the interior of the circle.
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26. What about (-5, -3)?
In the exterior of the circle.
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27. You could do this…
Find the distance from the center (-1, 1) to the point (-5, -3):
Since the distance to the point is larger than the radius, it must be in the exterior of the circle. 5
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28. What you can now do: Write the equation of a circle.
Graph a circle from its equation.
Determine where a point is in the interior, exterior, or on a circle.
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29. Quick Practice
Identify the center and the radius of the circle: (x + 2)2 + y2 = 9
Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle.
Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1
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30. Quick Practice
Identify the center and the radius of the circle: (x + 2)2 + y2 = 9
Center (-2, 0) Radius = 3
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31. Quick Practice
Find the equation of a circle if the center is (1, 2) and the point (3, 0) is on the circle.
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32. Quick Practice
Sketch the graph of the circle given by the equation (x - 1)2 + (y + 3)2 = 1
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33. Practice Problems
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