4th EDITION

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College Algebra Trigonometry and Precalculus
4th EDITION
2
Circles

• 2.2

Center-Radius Form General Form An Application
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Circle-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.

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A 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
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Example 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.
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Example 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
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Example 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
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Example 2

• GRAPHING CIRCLES

Graph the circle.
a.
Solution
Gives ( 3, 4) as the center and 6 as the radius.
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Example 2

• GRAPHING CIRCLES

y
Graph the circle.
a.
Solution
6
(x, y)
( 3, 4)
x
Gives ( 3, 4) as the center and 6 as the radius.
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Example 2

• GRAPHING CIRCLES

y
Graph the circle.
b.
Solution
(x, y)
3
x
Gives (0, 0) as the center and 3 as the radius.
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The equation
for some real numbers c, d, and e, can have a
graph that is a circle or a point, or is
nonexistent.
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General 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.

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Example 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
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Example 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.
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Example 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.
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Example 4

• FINDING THE CENTER AND RADIUS BY COMPLETING THE
  SQUARE

Group the terms factor out 2.
Be careful here.
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Example 4

• FINDING THE CENTER AND RADIUS BY COMPLETING THE
  SQUARE

Factor simplify on the right.
Divide both sides by 2.
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Example 4

• FINDING THE CENTER AND RADIUS BY COMPLETING THE
  SQUARE

Divide both sides by 2.
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Example 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
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Example 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.
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Example 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).)
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Example 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.
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Example 6

• LOCATING THE EPICENTER OF AN EARTHQUAKE

Station A
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Example 6

• LOCATING THE EPICENTER OF AN EARTHQUAKE

Station B
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Example 6

• LOCATING THE EPICENTER OF AN EARTHQUAKE

Station C
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Example 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).