SECT
ION
12.2
The Circle
The second conic section we look at is the circle. The circle can be described by using the standard form for a conic section,
12.2
12.2 O B J E C T I V E S
ax2 by2 cxy dx ey f 0
1. Identify the graph
of an equation as
a line, a parabola,
or a circle
2. Write the equation
of a circle in standard form and
graph the circle
but we will develop the standard form for a circle through the definition of a circle.
A circle is the set of all points in the plane equidistant from a fixed point, called
the center of the circle. The distance between the center of the circle and any
point on the circle is called the radius of the circle.
The distance formula is central to any discussion of conic sections.
The Distance Formula
y
(x2, y2)
The distance d between two points (x1, y1) and (x2, y2) is given by
2
2
d (x
2
x
(
y2
y
1)
1)
d
(x1, y1)
(x2, y1)
x
We can use the distance formula to derive the algebraic equation of a circle, given its
center and its radius.
Suppose a circle has its center at a point with coordinates (h, k) and radius r. If
(x, y) represents any point on the circle, then, by its definition, the distance from
(h, k) to (x, y) is r. Applying the distance formula, we have
r (x
h
)2
(y
k
)2
y
Squaring both sides of the equation gives the equation of the circle
r2 (x h)2 (y k)2
(x, y)
r
In general, we can write the following equation of a circle.
(h, k)
x
785
786
Chapter 12
Conic Sections
■
A special case is the circle
centered at the origin with
radius r. Then (h, k) (0, 0),
and its equation is
2
2
x y r
Equation of a Circle
The equation of a circle with center (h, k) and radius r is
2
(x h)2 (y k)2 r 2
(1)
Equation (1) can be used in two ways. Given the center and radius of the circle,
we can write its equation; or given its equation, we can find the center and radius of
a circle.
Finding the Equation of a Circle
Example 1
Find the equation of a circle with center at (2, 1) and radius 3. Sketch the circle.
Let (h, k) (2, 1) and r 3. Applying equation (1) yields
y
(x 2)2 [y (1)]2 32
(x 2)2 (y 1)2 9
3
x
(2, 1)
(x 2)2 (y 1)2 9
To sketch the circle, we locate the center of the circle. Then we determine four points
3 units to the right and left and up and down from the center of the circle. Drawing a
smooth curve through those four points completes the graph.
✓ CHECK YOURSELF 1
■
Find the equation of the circle with center at (2, 1) and radius 5. Sketch the circle.
Now, given an equation for a circle, we can also find the radius and center and
then sketch the circle. We start with an equation in the special form of equation (1).
Example 2
Finding the Center and Radius of a Circle
Find the center and radius of the circle with equation
(x 1)2 (y 2)2 9
Remember, the general form is
(x h)2 (y k)2 r 2
Section 12.2
■
The Circle
787
Our equation “fits” this form when it is written as






Note: y 2 y (2)
The circle can be graphed
on the calculator by
solving for y, then
graphing both the upper
half and lower half of the
circle. In this case,
(x 1)2 [y (2)]2 32
So the center is at (1, 2), and the radius is 3. The graph is shown.
y
(x 1)2 (y 2)2 9
(y 2)2 9 (x 1)2
(y 2) 9
(
x
1
)2
x
y 2 9
(x
1
)
2
3
Now graph the two
functions
(1, 2)
(x
1
)2
y 2 9
and
(x 1)2 (y 2)2 9
(x
1
)2
y 2 9
on your calculator. (The
display screen may need
to be squared to obtain
the shape of a circle.)
✓ CHECK YOURSELF 2
■
Find the center and radius of the circle with equation
(x 3)2 (y 2)2 16
Sketch the circle.
To graph the equation of a circle that is not in standard form, we complete the
square. Let’s see how completing the square can be used in graphing the equation of
a circle.
Example 3
To recognize the equation as
having the form of a circle,
note that the coefficients of x2
and y2 are equal.
Finding the Center and Radius of a Circle
Find the center and radius of the circle with equation
x2 2x y2 6y 1
Then sketch the circle.
The linear terms in x and y
show a translation of the
center away from the origin.
We could, of course, simply substitute values of x and try to find the corresponding
values for y. A much better approach is to rewrite the original equation so that it matches
the standard form.
788
Chapter 12
■
Conic Sections
First, add 1 to both sides to complete the square in x.
y
x2 2x 1 y2 6y 1 1
3
Then add 9 to both sides to complete the square in y.
(1, 3)
x2 2x 1 y2 6y 9 1 1 9
x
(x 1) (y 3) 9
2
2
We can factor the two trinomials on the left (they are both perfect squares) and simplify on the right.
(x 1)2 (y 3)2 9
The equation is now in standard form, and we can see that the center is at (1, 3) and
the radius is 3. The sketch of the circle is shown. Note the “translation” of the center
to (1, 3).
✓ CHECK YOURSELF 3
■
Find the center and radius of the circle with equation
x2 4x y2 2y 1
Sketch the circle.
✓ CHECK YOURSELF ANSWERS
■
1. (x 2)2 (y 1)2 25.
2. (x 3)2 (y 2)2 16.
y
y
4
5
(3, 2)
(2, 1)
x
3. (x 2)2 (y 1)2 4.
y
2
(2, 1)
x
E xercises
1. Parabola
2. Circle
3. Line
4. Line
5. Circle
6. Parabola
7. Circle
8. Line
9. None of These
10. Circle
11. Parabola
12. None of These
■
12.2
In Exercises 1 to 12, decide whether each equation has as its graph a line, a parabola,
a circle, or none of these.
1. y x2 2x 5
2. y2 x2 64
3. y 3x 2
4. 2y 3x 12
5. (x 3)2 (y 2)2 10
6. y 2(x 3)2 5
7. x2 4x y2 6y 3
8. 4x 3
13. Center: (0, 0); radius: 5
14. Center: (0, 0); radius: 62
15. Center: (3, 1); radius: 4
9. y2 4x2 36
11. y 2x2 8x 3
10. x2 (y 3)2 9
12. 2x2 3y2 6y 13
16. Center: (3, 0); radius: 9
17. Center: (1, 0); radius: 4
18. Center: (0, 3); radius: 9
19. Center: (3, 4);
radius: 4
1
5 3
20. Center: , ;
2 2
6
6
radius: 2
21. Center (0, 0); radius: 2
22. Center (0, 0); radius: 5
23. Center (0, 0); radius: 3
In Exercises 13 to 20, find the center and the radius for each circle.
13. x2 y2 25
14. x2 y2 72
15. (x 3)2 (y 1)2 16
16. (x 3)2 y2 81
17. x2 2x y2 15
18. x2 y2 6y 72
19. x2 6x y2 8y 16
20. x2 5x y2 3y 8
In Exercises 21 to 32, graph each circle by finding the center and the radius.
21. x2 y2 4
22. x2 y2 25
23. 4x2 4y2 36
24. 9x2 9y2 144
25. (x 1)2 y2 9
26. x2 (y 2)2 16
27. (x 4)2 (y 1)2 16
28. (x 3)2 (y 2)2 25
24. Center (0, 0); radius: 4
25. Center (1, 0); radius: 3
26. Center (0, 2); radius: 4
27. Center (4, 1); radius: 4
28. Center (3, 2); radius: 5
789
790
Chapter 12
■
Conic Sections
29. Center (0, 2); radius: 4
29. x2 y2 4y 12
30. x2 6x y2 0
30. Center (3, 0); radius: 3
31. x2 4x y2 2y 1
32. x2 2x y2 6y 6
31. Center (2, 1);
radius: 2
33. Describe the graph of x2 y2 2x 4y 5 0.
32. Center (1, 3);
radius: 4
34. Describe how completing the square is used in graphing circles.
33. Circle with radius zero
35. A solar oven is constructed in the shape of a hemisphere. If the equation
35. 105
cm; 22.4 cm
2
x2 y2 500
2
36. x y 1600
4
37. x2 y2 9
8
38. m
3
describes the circumference of the oven in centimeters, what is its radius?
36. A solar oven in the shape of a hemisphere is to have a diameter of 80 cm. Write
the equation that describes the circumference of this oven.
6
x2
39. y 3
y 3
6
x2
37. A solar water heater is constructed in the shape of a half cylinder, with the water
4
supply pipe at its center. If the water heater has a diameter of m, what is the
3
equation that describes its circumference?
40. y 9
(x
3
)2
y 9
(x
3
)2
38. A solar water heater is constructed in the shape of a half cylinder having a circumference described by the equation
41. y 3
6
(x
5
)2
y 3
6
(x
5
)2
42. y 1 2
5
(x
2
)2
y 1 2
5
(x
2
)2
9x2 9y2 16 0
What is its diameter if the units for the equation are meters?
A circle can be graphed on a calculator by plotting the upper and lower semicircles on
the same axes. For example, to graph x2 y2 16, we solve for y:
y 16
x2
43. Domain: 7 x 1;
range: 2 y 6
This is then graphed as two separate functions,
y 16
x2
44. Domain: 2 x 4;
range: 2 y 8
45. Domain: 5 x 5;
range: 2 y 8
46. Domain: 8 x 4;
range: 6 y 6
and
y 16
x2
In Exercises 39 to 42, use that technique to graph each circle.
39. x2 y2 36
40. (x 3)2 y2 9
41. (x 5)2 y2 36
42. (x 2)2 (y 1)2 25
Each of the following equations defines a relation. Write the domain and the range of
each relation.
43. (x 3)2 (y 2)2 16
44. (x 1)2 (y 5)2 9
45. x2 (y 3)2 25
46. (x 2)2 y2 36