17.1 Equation of a Circle

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Name
Class
Date
17.1 Equation of a Circle
Essential Question: How can you write the equation of a circle if you know its radius and the
coordinates of its center?
Resource
Locker
Explore
Deriving the Equation of a Circle
You have already worked with circles in several earlier lessons. Now you will investigate circles
in a coordinate plane and learn how to write an equation of a circle.
We can define a circle as the set of all points in the coordinate plane that are a fixed distance r
from the center (h, k).
Consider the circle in a coordinate plane that has its center at C(h, k) and that has radius r.
A
Let P be any point on the circle and let the coordinates of P be (x, y).
Create a right triangle by drawing a horizontal line through C and a vertical line through P,
as shown.
What are the coordinates of point A? A(x, k) Explain how you found the coordinates of A.
A has the same x-coordinate as P and the same y-coordinate as C.
y
© Houghton Mifflin Harcourt Publishing Company
P(x, y)
r
C(h, k)
A
x
0
B
Use absolute value to write expressions for the lengths of the legs of △CAP.
CA =
C
⎜x - h⎟ ; PA = ⎜y - k⎟
Use the Pythagorean Theorem to write a relationship among the side lengths of△CAP.
(x - h) + (y - k) 2 =
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Correcti
Reflect
1.
Compare your work with that of other students. Then, write an equation for the
circle with center C(h, k) and radius r.
2
2
(x - h) + (y - k) = r 2
2.
Why do you need absolute values when you write expressions for the lengths of
the legs in Step B, but not when you write the relationship among the side lengths in Step C?
In Step B, the absolute values ensure that the lengths are nonnegative.
In Step C, the lengths are squared, so the absolute values are no longer needed.
3.
Suppose a circle has its center at the origin. What is the equation of the circle in this case?
x2 + y2 = r2
Writing the Equation of a Circle
Explain 1
You can write the equation of a circle given its coordinates and radius.
Equation of a Circle
The equation of a circle with center (h, k) and radius r is (x - h) + (y - k) = r 2.
2
2
Example 1 Write the equation of the circle with the given center and radius.
 Center:
(-2, 5) ; radius: 3
(x - h) 2 + (y - k) 2 = r 2
(x - (-2))
2
+ (y - 5) =3 2
(4, -1);radius:
Simplify.
―
√5
( ( )) ( ( ))
2
4
+ y - -1
(x -4 ) + (y +1 ) =
2
(x - h) 2 + (y - k) 2 = r 2
x-
Substitute −2 for h, 5 for k, and 3 for r.
2
(x + 2) 2 + (y - 5) 2 = 9
 Center:
Write the general equation of a circle.
Write the general equation of a circle.
2
―
= √―
5 Substitute
2
4 for h, -1 for k, and √5 for r.
2
5
Simplify.
Reflect
4.
Suppose the circle with equation (x - 2) + (y + 4) = 7 is translated by (x, y) → (x + 3, y - 1).
What is the equation of the image of the circle? Explain.
The center of the circle, (2, -4) , is translated to(2 + 3, -4 - 1) = (5, -5),
2
2
and the radius does not change. Therefore, the image of the circle has
equation (x - 5) + (y + (-5)) = 7 or (x - 5) + (y + 5) = 7.
2
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Your Turn
Write the equation of the circle with the given center and radius.
5.
Center: (4, 3) ; radius: 4
6.
Center: (-1, -1);radius: √3
(x - 4) 2 + (y - 3) 2 = 16
(x + 1) + ( y + 1) = 3
2
2
―
Finding the Center and Radius of a Circle
Explain 2
Sometimes you may find an equation of a circle in a different form. In that case, you may need to rewrite the equation
to determine the circle’s center and radius. You can use the process of completing the square to do so.
Example 2 Find the center and radius of the circle with the given equation.
Then graph the circle.

x 2 - 4x + y 2 + 2y = 20
Step 1
Complete the square twice to write the equation in the form (x - h) + (y - k) = r 2.
2
x 2 - 4x + ( ) + y 2 + 2y + ( ) = 20 + ( )
2
( )
2
()
2
( ) ()
Set up to complete the square.
2 = 20 + _
-4 + _
2 Add
-4 +y 2 + 2y + _
x 2 - 4x + _
2
2
2
2
2
2
x 2 - 4x + 4 + y 2 + 2y + 1 = 20 + 5
2
(x - 2) 2 + (y + 1) 2 = 25
Step 2
2
2
-4 and _
( 22 )
(_
2 )
2
2
to both sides.
Simplify.
Factor.
Identify h, k, and r to determine the center and radius.
h=2
k = -1
―
r = √25 = 5
So, the center is (2, -1) and the radius is 5.
Step 3
Graph the circle.
5
y
• Locate the center of the circle.
• Place the point of your compass at the center
x
• Open the compass to the radius.
-5
• Use the compass to draw the circle.
0
5
-5
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x 2 + 6x + y 2 - 4y + 4 = 0
Step 1 Complete the square twice to write the equation in the form (x - h) + (y - k) = r 2.
2
x 2 + 6x +
( )
_6
2
2
+ y 2 - 4y +
( )
-4
___
2
2
= -4 +
( ) ( )
_6
2
2
+
-4
___
2
2
2
Add
( )
_6
2
2
and
to both sides.
( )
-4
___
2
2
Subtract 4 from both sides.
x 2 + 6x + 9 + y 2 - 4y + 4 = -4 + 13
(x +3 ) + (y -2 ) =
2
Simplify.
2
9
Factor.
Step 2 Identify h, k, and r to determine the center and radius.
_
h = -3
k= 2
(
So, the center is -3 , 2
Step 3 Graph the circle.
r=
)
√
9 = 3
and the radius is 3 .
5
• Locatethecenterofthecircle.
• Placethepointofyourcompassatthecenter
• Openthecompasstotheradius.
y
x
0
-5
• Usethecompasstodrawthecircle.
5
-5
Reflect
How can you check your graph by testing specific points from the graph in the
original equation? Give an example.
Check that points on the graph satisfy the equation. For Example 2A, (5, 3) is on the graph
and this point satisfies the equation: 5 2 - 4(5) + 3 2 + 2(3) = 20.
Your Turn
8.
Find the center and radius of the circle with the equation.
x 2 + 2x + y 2 - 8y + 13 = 0. Then graph the circle.
6
x 2 + 2x + 1 + y 2 - 8y + 16 = -13 + 1 + 16
4
(x + 1) 2 + (y - 4) 2 = 2 2
2
Center: (-1, 4), radius: 2
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9.
Find the center and radius of the circle with the equation
x 2 + y 2 + 4y = 5. Then graph the circle.
x 2 +y 2 + 4y + 4 = 5 + 4
(x - 0) 2 + (y + 2) 2 =3 2
2
y
x
0
-4
4
-2
Center (0, −2) , radius 3
-4
Writing a Coordinate Proof
Explain 3
You can use a coordinate proof to determine whether or not a given point lies on a given circle in
the coordinate plane.
Example 3 Prove or disprove that the given point lies on the given circle.
 Point
Step1
_
(3, √7 )
circle centered at the origin and containing the point(-4, 0)
Plot a point at the origin and at(-4, 0) . Use these to help you draw the circle
centered at the origin that contains (-4, 0).
5
y
x
-5
0
5
-5
Step 2
Determine the radius: r = 4
Step 3
Use the radius and the coordinates of the center to write the equation of the circle.
(x - h) 2 + (y - k) 2 = r 2
Step 4
Write the equation of the circle.
(x - 0) 2 + (y - 0) 2 = (4) 2
Substitute 0 for h, 0 for k, and 4 for r.
x 2 + y 2 = 16
Simplify.
_
Substitute the x- and y-coordinates of the point (3, √7 ) in the equation of the circle to check whether
they satisfy the equation of the circle.
_
(3) 2 + (√7 ) ≟ 16
2
9 + 7 = 16
_
Substitute 3 for x and √7for y.
Simplify.
_
So, the point(3,√7 ) lies on the circle because the point’sx- and y-coordinates satisfy the equation
of the circle.
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―
DO NOT
Correcti
Point (1, √6 ), circle with center (−1, 0) and containing the point (−1, 3)
(
)
(
Step 1 Plot a point at -1 , 0 and at -1, 3
(
contains -1, 3
)
)
(
)
. Draw the circle centered at -1 , 0 that
.
5
y
x
-5
0
5
-5
Step 2 Determine the radius: r = 3
Step 3 Use the radius and the coordinates of the center to write the equation of the circle.
(x - h)2 + (y - k)2 = r 2
Write the equation of the circle.
( ( )) ( ( )) ( )
x - -1
2
2
+ y-
(x + 1 ) + y =
=
0
2
3
Substitute -1 for h, 0 for k, and 3 for r.
2
2
9
Simplify.
_
Step 4 Substitute the x- and y-coordinates of the point (1, √6 ) in the equation of the circle to check whether
they satisfy the equation of the circle.
1+1
2
4 + 6 ≠ 9 ⊣
―
9
―
Substitute 1 for x and √6 for y.
Simplify.
So the point (1, √6 ) does not lie on the circle because the x- and y-coordinates of the
point
do not satisfy
the equation of the circle.
Reflect
10. How do you know that the radius of the circle in Example 3A is 4?
The line segment joining the center, (0, 0), to point (−4, 0) is a radius of the circle, and
(
) + (√―6 ) ≟
2
since it is horizontal, you can count squares to determine the radius.
―
―
11. Name another point with noninteger coordinates that lies on the circle in Example 3A. Explain.
2
Possible answer: (3, √7 ), because 3 2 + ( √7 ) = 16.
Module 17
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Lesson 1
24/03/14 10:02 AM
Your Turn
Prove or disprove that the given point lies on the given circle.
―
12. Point ( √18 , -4), circle centered at the origin and containing the point (6, 0)
r=6
(x - 1)2 + (y - 0)2 = (6)2
( √―
18 )
2
x 2 + y 2 = 36
+ (-4) ≟ 36
2
18 + 16 ≠ 36
So the point ( √18 , -4) does not lie on the circle because the point’s x- and y-coordinates
do not satisfy the equation of the circle.
―
13. Point (4, −4), circle with center (1, 0) and containing the point (1, 5)
r=5
(x - 1)2 + (y - 0)2 = (5)2
(x - 1)2 + y 2 = 25
(4 - 1)2 + (-4)2 ≟ 25
9 + 16 = 25
So the point (4, −4) lies on the circle because the point’s x- and y-coordinates satisfy the
equation of the circle.
Elaborate
14. Discussion How is the distance formula related to the equation of a circle?
Possible answer: Given the points (x, y) and (h, k) and the distance formula. You get
d=
――――――
√(x - h) + (y - k) . But if you square both sides, you get d
2
2
2
= (x - h) + (y - k) ,
2
2
which is the same as the equation of a circle centered at the origin with radius d.
15. Essential Question Check-In What information do you need to know to write
the equation of a circle?
You need the coordinates of the center and the radius.
Module 17
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Lesson 1
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17.1 Equation of a Circle

Transcription

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