10-2
10-2 Circles
Circles
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra2
Holt
Algebra
2
10-2
Circles
Warm Up
Find the slope of the line that connects
each pair of points.
1. (5, 7) and (–1, 6)
2. (3, –4) and (–4, 3)
Holt Algebra 2
1
6
–1
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Circles
Warm Up
Find the distance between each pair of
points.
3. (–2, 12) and (6, –3)
4. (1, 5) and (4, 1)
Holt Algebra 2
17
5
10-2
Circles
Objectives
Write an equation for a circle.
Graph a circle, and identify its center
and radius.
Holt Algebra 2
10-2
Circles
Vocabulary
circle
tangent
Holt Algebra 2
10-2
Circles
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.
Holt Algebra 2
10-2
Circles
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.
Holt Algebra 2
10-2
Circles
Check It Out! Example 1
Write the equation of a circle with center (4, 2)
and radius r = 7.
Use the Distance Formula with (x2, y2) = (x, y),
(x1, y1) = (4, 2), and distance equal to the radius, 7.
Use the Distance Formula.
Substitute.
Square both sides.
Holt Algebra 2
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Circles
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.
Holt Algebra 2
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Circles
Helpful Hint
If the center of the circle is at the origin, the
equation simplifies to x2 + y2 = r2.
Holt Algebra 2
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Circles
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
Holt Algebra 2
10-2
Circles
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.
(x + 4)2 + (y – 11)2 = 152
(x + 4)2 + (y – 11)2 = 225
Holt Algebra 2
Substitute the values into the
equation of a circle.
10-2
Circles
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.
(x + 3)2 + (y – 5)2 = 132
2
2
(x + 3) + (y – 5) = 169
Holt Algebra 2
Substitute the values into
the equation of a circle.
10-2
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.
Holt Algebra 2
10-2
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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 < 42. Points B, C, D and E
are within a 4-mile radius .
Check Point F(–2, –3) is near the boundary.
2
2
2
(–2 + 1) + (–3 – 1) < 4
(–1)2 + (–4)2 < 42
1 + 16 < 16 x Point F (–2, –3) is not inside the circle.
Holt Algebra 2
10-2
Circles
Check It Out! Example 3
What if…? Which homes are within a 3-mile
radius of a restaurant located at (2, –1)?
The circle has a center (2, –1) and
radius 3. The points inside the circle
2
will satisfy the inequality (x – 2) +
(y + 1)2 < 32. Points C and E are
within a 3-mile radius .
Check Point B (1, 2) is near the boundary.
(1 – 2)2 + (2 + 1)2 < 32
(–1)2 + (3)2 < 32
Point B (1, 2) is not inside the circle.
1+9<9 x
Holt Algebra 2
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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.
Remember!
To review linear functions, see Lesson 2-4.
Holt Algebra 2
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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.
From the equation x2 + y2 = 29, the circle has
center of (0, 0) and radius r =
.
Holt Algebra 2
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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 ).
The slope of the radius is
5
2
.
Because the slopes of perpendicular lines are
negative reciprocals, the slope of the tangent is –
Holt Algebra 2
2
5
.
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Example 4 Continued
Step 3 Find the slope-intercept equation of the
tangent by using the point (2, 5) and the slope
m = – 25 .
Use the point-slope formula.
2
Substitute (2, 5) (x1 , y1 ) and – 5 for m.
Rewrite in slope-intercept form.
Holt Algebra 2
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Example 4 Continued
The equation of the line that is tangent to
x2 + y2 = 29 at (2, 5) is
.
Check Graph the
circle and the line.
Holt Algebra 2
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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.
Holt Algebra 2
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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 ).
The slope of the radius is
–3
4
.
Because the slopes of perpendicular lines are
negative reciprocals, the slope of the tangent is
Holt Algebra 2
.
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Check It Out! Example 4 Continued
Step 3. Find the slope-intercept equation of the
tangent by using the point (5, –5) and the slope
.
Use the point-slope formula.
Substitute (5, –5 ) for (x1 , y1 ) and
for m.
Rewrite in slope-intercept form.
Holt Algebra 2
4
3
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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.
Holt Algebra 2
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Circles
Lesson Quiz: Part I
1. Write an equation for the circle with
center (1, –5) and a radius of
.
2
2
(x – 1) + (y + 5) = 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
Holt Algebra 2
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Lesson Quiz: Part II
3. Which points on the graph shown are
within 2 units of the point (0, –2.5)?
C, F
Holt Algebra 2
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Circles
Lesson Quiz: Part III
4. Write an equation for the line tangent to
2
2
the circle x + y = 17 at the point (4, 1).
y – 1 = –4(x – 4)
Holt Algebra 2