The document provides examples and explanations of completing the square to derive the standard form of a circle equation. It defines a circle as all points equidistant from a center point, and discusses radius, diameter, and graphing circles. Examples demonstrate finding the equation, center, and radius of circles given algebraic or geometric information. The general form of a circle equation is derived.

1. Circles
Mathematics 4
August 10, 2011
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2. Review of Completing the Square
Completing the Square
Express the following quadratic expressions as (y − k) = a(x − h)2 .
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3. Review of Completing the Square
Completing the Square
Express the following quadratic expressions as (y − k) = a(x − h)2 .
1 x2 − y − 12x + 7 = 0
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4. Review of Completing the Square
Completing the Square
Express the following quadratic expressions as (y − k) = a(x − h)2 .
1 x2 − y − 12x + 7 = 0
→ (y + 29) = (x − 6)2
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5. Review of Completing the Square
Completing the Square
Express the following quadratic expressions as (y − k) = a(x − h)2 .
1 x2 − y − 12x + 7 = 0
→ (y + 29) = (x − 6)2
2 2x2 − 5x − y − 3 = 0
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6. Review of Completing the Square
Completing the Square
Express the following quadratic expressions as (y − k) = a(x − h)2 .
1 x2 − y − 12x + 7 = 0
→ (y + 29) = (x − 6)2
2 2x2 − 5x − y − 3 = 0
49
→y+ 8 = 2(x − 5 )2
4
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7. Review of Completing the Square
Completing the Square
Express the following expressions as (x − h)2 + (y − k)2 = r, where h, k,
and r are constants.
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8. Review of Completing the Square
Completing the Square
Express the following expressions as (x − h)2 + (y − k)2 = r, where h, k,
and r are constants.
1 x2 + y 2 + 2x − 8y + 4 = 0
Mathematics 4 () Circles August 10, 2011 3 / 17

9. Review of Completing the Square
Completing the Square
Express the following expressions as (x − h)2 + (y − k)2 = r, where h, k,
and r are constants.
1 x2 + y 2 + 2x − 8y + 4 = 0
→ (x + 1)2 + (y − 4)2 = 13
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10. Review of Completing the Square
Completing the Square
Express the following expressions as (x − h)2 + (y − k)2 = r, where h, k,
and r are constants.
1 x2 + y 2 + 2x − 8y + 4 = 0
→ (x + 1)2 + (y − 4)2 = 13
2 9x2 + 9y 2 + 6x − 12y + 5 = 63
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11. Review of Completing the Square
Completing the Square
Express the following expressions as (x − h)2 + (y − k)2 = r, where h, k,
and r are constants.
1 x2 + y 2 + 2x − 8y + 4 = 0
→ (x + 1)2 + (y − 4)2 = 13
2 9x2 + 9y 2 + 6x − 12y + 5 = 63
→ (x + 3 )2 + (y − 2 )2 = 7
1
3
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12. Circles
What is a circle?
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13. Circles
Definition of Circles
A circle is a set of all points (locus) that are the same distance from a
given point.
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14. Circles
Definition of Circles
A circle is a set of all points (locus) that are the same distance from a
given point.
Terminology
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15. Circles
Definition of Circles
A circle is a set of all points (locus) that are the same distance from a
given point.
Terminology
same distance → radius
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16. Circles
Definition of Circles
A circle is a set of all points (locus) that are the same distance from a
given point.
Terminology
same distance → radius
given point → center
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17. The Standard Form of the Circle Equation
The Distance Formula
The distance between two points (x1 , y1 ) and (x2 , y2 ) in the Cartesian
plane is given by:
d= (x2 − x1 )2 + (y2 − y1 )2
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18. The Standard Form of the Circle Equation
The Distance Formula
Use the distance formula to relate the radius with the center of the circle.
r= (x − h)2 + (y − k)2 (1)
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19. The Standard Form of the Circle Equation
Standard Form/Center-Radius Form
Given a circle with center at (h, k) and having a radius r, the center radius
form of the circle equation is given by:
(x − h)2 + (y − k)2 = r2
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20. Graphing Examples
Graph x2 + y 2 = 10. Label center, radius, and any intercepts.
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21. Graphing Examples
Graph x2 + y 2 = 10. Label center, radius, and any intercepts.
√ √
x-intercepts → 10, − 10
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22. Graphing Examples
Graph x2 + y 2 = 10. Label center, radius, and any intercepts.
√√
x-intercepts → 10, − 10
√ √
y-intercepts → 10, − 10
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23. Graphing Examples
Graph (x + 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts.
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24. Graphing Examples
Graph (x + 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts.
√ √
x-intercepts → −3 + 5, −3 − 5
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25. Graphing Examples
Graph (x + 3)2 + (y − 2)2 = 9. Label center, radius, and any intercepts.
√ √
x-intercepts → −3 + 5, −3 − 5
y-intercepts → 2
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26. Problems on Circles
Example 1
What is the equation of a circle with radius 5, centered on the origin?
Graph this circle.
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27. Problems on Circles
Example 1
What is the equation of a circle with radius 5, centered on the origin?
Graph this circle.
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28. Problems on Circles
Example 1
What is the equation of a circle with radius 5, centered on the origin?
Graph this circle.
x2 + y 2 = 25
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29. Problems on Circles
Example 1
What is the equation of a circle with radius 5, centered on the origin?
Graph this circle.
x2 + y 2 = 25
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30. Problems on Circles
Example 2
Move the circle in the previous problem 3 units to the left and 2 units up.
What is its equation? Graph this circle and note the intercepts.
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31. Problems on Circles
Example 2
Move the circle in the previous problem 3 units to the left and 2 units up.
What is its equation? Graph this circle and note the intercepts.
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32. Problems on Circles
Example 2
Move the circle in the previous problem 3 units to the left and 2 units up.
What is its equation? Graph this circle and note the intercepts.
(x + 3)2 + (y − 2)2 = 25
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33. Problems on Circles
Example 2
Move the circle in the previous problem 3 units to the left and 2 units up.
What is its equation? Graph this circle and note the intercepts.
(x + 3)2 + (y − 2)2 = 25
√
x-intercepts : −3 ± 21, y-intercepts: 6, −2
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34. Problems on Circles
Example 3
Find an equation of a circle with a diameter whose endpoints are at
P1 (7, −3) and P2 (1, 7).
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35. Problems on Circles
Example 3
Find an equation of a circle with a diameter whose endpoints are at
P1 (7, −3) and P2 (1, 7).
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36. Problems on Circles
Example 3
Find an equation of a circle with a diameter whose endpoints are at
P1 (7, −3) and P2 (1, 7).
center: Use the midpoint formula
x1 + x2 y1 + y2
→ (h, k) = , = (4, 2)
2 2
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37. Problems on Circles
Example 3
Find an equation of a circle with a diameter whose endpoints are at
P1 (7, −3) and P2 (1, 7).
center: Use the midpoint formula
x1 + x2 y1 + y2
→ (h, k) = , = (4, 2)
2 2
radius: Distance from one endpoint to the center
√
→ r = (x1 − h)2 + (y1 − k)2 = 34
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38. Problems on Circles
Example 3
Find an equation of a circle with a diameter whose endpoints are at
P1 (7, −3) and P2 (1, 7).
center: Use the midpoint formula
x1 + x2 y1 + y2
→ (h, k) = , = (4, 2)
2 2
radius: Distance from one endpoint to the center
√
→ r = (x1 − h)2 + (y1 − k)2 = 34
(x − 4)2 + (y − 2)2 = 34
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39. The General Form of the Circle Equation
Rewriting the answer to the previous problem:
(x − 4)2 + (y − 2)2 = 34
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40. The General Form of the Circle Equation
Rewriting the answer to the previous problem:
(x − 4)2 + (y − 2)2 = 34 → x2 + y 2 − 8x − 4y − 14 = 0
This is called the General Form of the Circle Equation.
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41. The General Form of the Circle Equation
The General Form of the Circle Equation
Ax2 + Ay 2 + Cx + Dy + E = 0
The x2 and y 2 terms should have identical coefficients.
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42. Problems on Circles
Example 4
Find the center and radius of the circle with the equation
x2 + y 2 + 4x − 6y + 5 = 0.
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43. Problems on Circles
Example 4
Find the center and radius of the circle with the equation
x2 + y 2 + 4x − 6y + 5 = 0.
(x2 + 4x) + (y 2 − 6y) = −5
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44. Problems on Circles
Example 4
Find the center and radius of the circle with the equation
x2 + y 2 + 4x − 6y + 5 = 0.
(x2 + 4x) + (y 2 − 6y) = −5
(x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9
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45. Problems on Circles
Example 4
Find the center and radius of the circle with the equation
x2 + y 2 + 4x − 6y + 5 = 0.
(x2 + 4x) + (y 2 − 6y) = −5
(x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9
(x + 2)2 + (y − 3)2 = 8
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46. Problems on Circles
Example 4
Find the center and radius of the circle with the equation
x2 + y 2 + 4x − 6y + 5 = 0.
(x2 + 4x) + (y 2 − 6y) = −5
(x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9
(x + 2)2 + (y − 3)2 = 8
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47. Problems on Circles
Example 4
Find the center and radius of the circle with the equation
x2 + y 2 + 4x − 6y + 5 = 0.
(x2 + 4x) + (y 2 − 6y) = −5
(x2 + 4x+4) + (y 2 − 6y+9) = −5+4 + 9
(x + 2)2 + (y − 3)2 = 8
C(−2, 3)√
√
r= 8=2 2
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48. Problems on Circles
More examples
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49. Problems on Circles
More examples
1 Find the standard equation of a circle with center at (1, 5) and passes
through (7, 2).
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50. Problems on Circles
More examples
1 Find the standard equation of a circle with center at (1, 5) and passes
through (7, 2).
2 Find the area of the circle with equation x2 + y 2 + 8x − 12y − 14.
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51. Problems on Circles
More examples
1 Find the standard equation of a circle with center at (1, 5) and passes
through (7, 2).
2 Find the area of the circle with equation x2 + y 2 + 8x − 12y − 14.
3 Find the general equation of the circle tangent to both axes, whose
center is in QII, and whose radius is 4.
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