Write an equation for a circle.

1
Objectives
Write an equation for a circle. Graph a circle,
and identify its center and radius.
Vocabulary
circle tangent
2
Notes
1. Write an equation for the circle with center
(1, 5) and a radius of .
2. Write an equation for the circle with center
(4, 4) and containing the point (1, 16).
3. Write an equation for the line tangent to the
circle x2 y2 17 at the point (4, 1).
4. Which points on the graph shown are within 2
units of the point (0, 2.5)?
3
A circle is the set of points in a plane that are
a fixed distance, called the radius, from a fixed
point, called the center. Because all of the
points on a circle are the same distance from the
center of the circle, you can use the Distance
Formula to find the equation of a circle.
4
Example 1 Using the Distance Formula to Write
the Equation of a Circle
Write the equation of a circle with center (3,
4) and radius r 6.
Use the Distance Formula with (x2, y2) (x, y),
(x1, y1) (3, 4), and distance equal to the
radius, 6.
Use the Distance Formula.
Substitute.
Square both sides.
5
Notice that r2 and the center are visible in the
equation of a circle. This leads to the formula
for a circle with center (h, k) and radius r.
6
Example 2A Writing the Equation of a Circle
Write the equation of the circle.
the circle with center (0, 6) and radius r 1
(x h)2 (y k)2 r2
Equation of a circle
(x 0)2 (y 6)2 12
Substitute.
x2 (y 6)2 1
7
Example 2B Writing the Equation of a Circle
Write the equation of the circle.
the circle with center (4, 11) and containing
the point (5, 1)
Use the Distance Formula to find the radius.
Substitute the values into the equation of a
circle.
(x 4)2 (y 11)2 152
(x 4)2 (y 11)2 225
8
The location of points in relation to a circle
can be described by inequalities. The points
inside the circle satisfy the inequality (x h)2
(x k)2 lt r2. The points outside the circle
satisfy the inequality (x h)2 (x k)2 gt r2.
9
Example 3 Consumer Application
Use the map and information given in Example 3 on
page 730. Which homes are within 4 miles of a
restaurant located at (1, 1)?
The circle has a center (1, 1) and radius 4. The
points insides the circle will satisfy the
inequality (x 1)2 (y 1)2 lt 42. Points B, C,
D and E are within a 4-mile radius .
Check Point F(2, 3) is near the boundary.
(2 1)2 (3 1)2 lt 42
(1)2 (4)2 lt 42
x
Point F (2, 3) is not inside the circle.
1 16 lt 16
10
A tangent is a line in the same plane as the
circle that intersects the circle at exactly one
point. Recall from geometry that a tangent to a
circle is perpendicular to the radius at the
point of tangency.
11
Example 4 Writing the Equation of a Tangent
Write the equation of the line tangent to the
circle x2 y2 29 at the point (2, 5).
Step 1 Identify the center and radius of the
circle.
12
Example 4 Continued
Step 2 Find the slope of the radius at the point
of tangency and the slope of the tangent.
Use the slope formula.
Substitute (2, 5) for (x2 , y2 ) and (0, 0) for
(x1 , y1 ).
13
Example 4 Continued
Use the point-slope formula.
Rewrite in slope-intercept form.
14
Example 4 Continued
Check Graph the circle and the line.
15
Notes
1. Write an equation for the circle with center
(1, 5) and a radius of .
(x 1)2 (y 5)2 10
2. Write an equation for the circle with center
(4, 4) and containing the point (1, 16).
(x 4)2 (y 4)2 153
3. Write an equation for the line tangent to the
circle x2 y2 17 at the point (4, 1).
y 1 4(x 4)
16
Notes
4. Which points on the graph shown are within 2
units of the point (0, 2.5)?
C, F
17
Circles Extra Info
The following power-point slides contain extra
examples and information.
Reminder of Lesson Objectives Write an equation
for a circle. Graph a circle, and identify its
center and radius.
18
Write an equation for a circle. Graph a circle,
and identify its center and radius.
Vocabulary
circle tangent
19
Check It Out! Example 2
Find the equation of the circle with center (3,
5) and containing the point (9, 10).
Use the Distance Formula to find the radius.
Substitute the values into the equation of a
circle.
(x 3)2 (y 5)2 132
(x 3)2 (y 5)2 169
20
Check It Out! Example 4
Write the equation of the line that is tangent to
the circle 25 (x 1)2 (y 2)2, at the
point (1, 2).
Step 1 Identify the center and radius of the
circle.
From the equation 25 (x 1)2 (y 2)2, the
circle has center of (1, 2) and radius r 5.
21
Check It Out! Example 4 Continued
Step 2 Find the slope of the radius at the point
of tangency and the slope of the tangent.
Use the slope formula.
Substitute (5, 5) for (x2 , y2 ) and (1, 2) for
(x1 , y1 ).
22
Check It Out! Example 4 Continued
Use the point-slope formula.
Rewrite in slope-intercept form.
23
Check It Out! Example 4 Continued
The equation of the line that is tangent to 25
(x 1)2 (y 2)2 at (5, 5) is
.
Check Graph the circle and the line.