Title: 4th EDITION
1College Algebra Trigonometry and Precalculus
4th EDITION
2Circles
Center-Radius Form General Form An Application
3Circle-Radius form
- By definition, a circle is the set of all points
in a plane that lie a given distance from a given
point. The given distance is the radius of the
circle, and the given point is the center.
4A circle with center (h, k) and radius r has
equation
which is the center-radius form of the equation
of the circle. A circle with center (0, 0) and
radius r has equation
5Example 1
- FINDING THE CENTER-RADIUS FORM
Find the center-radius form of a circle with a
center at ( 3, 4), radius 6.
Solution
a. Use (h, k) ( 3, 4) and r 6
Center-radius form
Substitute
Watch signs here.
6Example 1
- FINDING THE CENTER-RADIUS FORM
Find the center-radius form of a circle with a
center at ( 3, 4), radius 6.
a. Use (h, k) ( 3, 4) and r 6
Solution
Substitute
7Example 1
- FINDING THE CENTER-RADIUS FORM
b. Find the center-radius form of a circle with
a center at (0, 0), radius 3.
Solution
Because the center is the origin and r 3, the
equation is
8Example 2
Graph the circle.
a.
Solution
Gives ( 3, 4) as the center and 6 as the radius.
9Example 2
y
Graph the circle.
a.
Solution
6
(x, y)
( 3, 4)
x
Gives ( 3, 4) as the center and 6 as the radius.
10Example 2
y
Graph the circle.
b.
Solution
(x, y)
3
x
Gives (0, 0) as the center and 3 as the radius.
11The equation
for some real numbers c, d, and e, can have a
graph that is a circle or a point, or is
nonexistent.
12General Form of the Equation of a Circle
- Consider
- There are three possibilities for the graph based
on the value of m. - If m gt 0, then r 2 m, and the graph of the
equation is a circle with the radius - If m 0, then the graph of the equation is the
single point (h, k). - If m lt 0, then no points satisfy the equation and
the graph is nonexistent.
13Example 3
- FINDING THE CENTER AND RADIUS BY COMPLETING THE
SQUARE
Show that x2 6x y2 10y 25 0 has a circle
as a graph. Find the center and radius.
Solution We complete the square twice, once for
x and once for y.
and
14Example 3
- FINDING THE CENTER AND RADIUS BY COMPLETING THE
SQUARE
Add 9 and 25 on the left to complete the two
squares, and to compensate, add 9 and 25 on the
right.
Complete the square.
Factor
Add 9 and 25 on both sides.
Since 9 gt 0, the equation represents a circle
with center at (3, 5) and radius 3.
15Example 4
- FINDING THE CENTER AND RADIUS BY COMPLETING THE
SQUARE
Show that 2x2 2y2 6x 10y 1 has a circle as
a graph. Find the center and radius.
Solution To complete the square, the
coefficients of the x2- and y2-terms must be 1.
Group the terms factor out 2.
16Example 4
- FINDING THE CENTER AND RADIUS BY COMPLETING THE
SQUARE
Group the terms factor out 2.
Be careful here.
17Example 4
- FINDING THE CENTER AND RADIUS BY COMPLETING THE
SQUARE
Factor simplify on the right.
Divide both sides by 2.
18Example 4
- FINDING THE CENTER AND RADIUS BY COMPLETING THE
SQUARE
Divide both sides by 2.
19Example 5
- DETERMINING WHETHER A GRAPH IS A POINT OR
NONEXISTENT
The graph of the equation x2 10x y2 4y 33
0 is either a point or is nonexistent. Which
is it?
Solution We complete the square for x and y.
Subtract 33.
and
20Example 5
- DETERMINING WHETHER A GRAPH IS A POINT OR
NONEXISTENT
The graph of the equation x2 10x y2 4y 33
0 is either a point or is nonexistent. Which
is it?
and
Complete the square.
Factor add.
21Example 5
- DETERMINING WHETHER A GRAPH IS A POINT OR
NONEXISTENT
Since 4 lt 0, there are no ordered pairs (x, y),
with both x and y both real numbers, satisfying
the equation. The graph of the given equation is
nonexistent it contains no points. ( If the
constant on the right side were 0, the graph
would consist of the single point ( 5, 2).)
22Example 6
- LOCATING THE EPICENTER OF AN EARTHQUAKE
Three receiving stations are located on a
coordinate plane at points (1, 4), ( 3, 1),
and (5, 2). The distance from the earthquake
epicenter to each station should be 2 units, 5
units, and 4 units respectively.
Solution Graph the three circles. From the
graph it appears that the epicenter is located at
(1, 2). To check this algebraically, determine
the equation for each circle and substitute x 1
and y 2.
23Example 6
- LOCATING THE EPICENTER OF AN EARTHQUAKE
Station A
24Example 6
- LOCATING THE EPICENTER OF AN EARTHQUAKE
Station B
25Example 6
- LOCATING THE EPICENTER OF AN EARTHQUAKE
Station C
26Example 6
- LOCATING THE EPICENTER OF AN EARTHQUAKE
Three receiving stations are located on a
coordinate plane at points (1, 4), ( 3, 1),
and (5, 2). The distance from the earthquake
epicenter to each station should be 2 units, 5
units, and 4 units respectively.
The point (1, 2) does lie on all three graphs
thus, we can conclude that the epicenter is at
(1, 2).