CHAPTER 12
Circles
OBJECTIVES
Write an equation for a circle.
Graph a circle, and identify its center and
radius.
CIRCLE

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.
FORMULA
CIRCLE
TO
WRITE
THE
EQUATION
OF A
Write the equation of a circle with center
(–3, 4) and radius r = 6.
 Solution:
 Use the Distance Formula with (x2, y2) = (x, y),
(x1, y1) = (–3, 4), and distance equal to the radius,
6.

EXAMPLE
Write the equation of a circle with center (4,
2) and radius r = 7.
 Solution:

CIRCLE

Notice that r2 and the center are visible in the
equation of a circle. This leads to a general
formula for a circle with center (h, k) and radius
r.
EXAMPLE 2A: WRITING
OF A CIRCLE
THE
EQUATION
Write the equation of the circle.
 the circle with center (0, 6) and radius r = 1
 Solution:
 (x – h)2 + (y – k)2 = r2
 (x – 0)2 + (y – 6)2 = 12
 x2 + (y – 6)2 = 1

EXAMPLE 2B: WRITING
OF A CIRCLE
THE
EQUATION
Write the equation of the circle.
 the circle with center (–4, 11) and
containing the point (5, –1)
 Solution
 Step 1:Use the distance formula

Step 2: Use the equation of the circle
2
2
2
 (x + 4) + (y – 11) = 15
2
2
 (x + 4) + (y – 11) = 225

CIRCLES

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 < r2.
The points outside the circle satisfy the
inequality (x – h)2 + (x – k)2 > r2.
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)?
TANGENT

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.
EXAMPLE





Write the equation of the line tangent to the
circle x2 + y2 = 29 at the point (2, 5).
Solution:
Step 1 Identify the center and radius of the
circle.
From the equation x2 + y2 = 29, the circle has
center of (0, 0) and radius r = .
Step 2 Find the slope of the radius at the point
of tangency and the slope of the tangent.
SOLUTION


Step 3 Find the slope-intercept equation of the
tangent by using the point (2, 5) and the slope
m = 5/2 .
SOLUTION

The equation of the line that is tangent to
2
y = 29 at (2, 5) is
x2 +
EXAMPLE

Write the equation of the line that is
2
2
tangent to the circle 25 = (x – 1) + (y + 2) ,
at the point (1, –2).
STUDENT GUIDED PRACTICE

Do even problems 2-10 in your book page 826
HOMEWORK

Do 12-20 in your book page 826
VIDEOS

Let’s Watch some videos
CLOSURE
Today we learned about circles
 Next class we are going to learn about ellipses
