2.2 parallel and perpendicular lines and circles slopes and parallel lines 1. if two nonvertical...
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2.2 Parallel and Perpendicular Lines and Circles
Slopes and Parallel Lines1. If two nonvertical lines are parallel,
then they have the same slopes.2. If two distinct nonvertical lines have
the same slope, then they are parallel.
3. Two distinct vertical lines, with undefined slopes, are parallel.
Example 1: Writing Equation of a Line Parallel to a Given Line
Write an equation of the line passing
through ( 2, 7) and parallel to the
line whose equation is 5 4.
Express the equation in slope-
intercept form.
y x
Solution
1 1
Notice that the line passes through the
point ( 2, 7). Using the point-slope
form of the line's equation, we have
2 and 7.x y
1 1( )y y m x x
Y1=-7 X1=-2
What is the slope of the line?
Given equation
5 4y x
Slope of the line is –5.
Parallel lines have the same slope.
So 5.m
X1=-2, y1=-7, and m=-5
( 7) 5( ( 2))y x 7 5 10y x
5 17y x This is the slope-intercept form of the
equation.
Practice Exercise
Write an equation of the line passing
through ( 1, 3) and parallel to the
line whose equation is 3 2 5 0.
Express the equation in slope-
intercept form.
x y
Answer to the Practice Exercise
3 9
2 2y x
Slopes and Perpendicular Lines1. If two nonvertical lines are
perpendicular, then the product of their slopes is –1.
2. If the product of the slopes of two lines is –1, then lines are perpendicular.
3. A horizontal line having zero slope is perpendicular to a vertical line having undefined slope.
Example 2: Finding the Slope of a Line Perpendicular to aGiven Line
Find the slope of any line that is
perpendicular to the line whose
equation is 3 2 6 0.x y
Solution Solve the given equation for y.
3 2 6 0x y 2 3 6y x
32 3y x
Slope is –3/2.
Given line has slope –3/2.
32
Any line perpendicular to this line has
a slope that is the negative reciprocal
of .Thus, the slope of any perpendicular
2line is .
3
Practice Exercise
The equation of a line is given by
3 4 7 0. Find the slope
of a line that is
(a) parallel to the line; and
(b) perpendicular to the line.
x y
Answers
3(a)
4
4(b)
3
Definition of a Circle
A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed distance from the circle’s center to any point on the circle is called the radius.
The Standard Form of the Equation of a Circle
2 2 2
The
with center (
standard form
,
of the equation of
) and radius
a
c isircle
( .) ( )
h
h y r
k
k
r
x
Center
Any point on the circle
Example 3 Finding the Standard Form of a Circle’s Equation
Write the standard form of the
equation of the circle with
center (2, 1) and radius 4.
Solution
Center ( , ) (2, 1)h k Radius 4r
2 2 2( ) ( )x h y k r 22 2( 2) ( 1) 4x y
2 2( 2) ( 1) 16x y
Practice Exercises
Write the standard form of the
equation of the circle with the given
center and radius.
1. Center (0,0), 8
2. Center ( 3,5), 3
r
r
Answers
2 2
2 2
1. 64
2. ( 3) ( 5) 9
x y
x y
Example 4: Using the Standard Form of a Circle’s
Equation to Graph the Circle
2 2
Find the center and radius of
the circle whose equation is
( 1) ( 4) 25
and graph the equation.
x y
Solution
Center ( 1,4)
Radius 5
Practice Exercise
2 2
Give the center and radius of the
circle described by the equation
( 4) ( 5) 36 and
graph the equation.
x y
Answer
Center ( 4, 5)
Radius 6
The General Form of the Equation of a Circle
2 2
general form of the
equation of a circle i
Th
s
e
.0x y Dx Ey F
Example 5: Converting the General Form of Circle’s
Equation to Standard Form and
Graphing the Circle
2 2
Write in standard form and graph:
8 4 16 0.x y x y
Solution2 2 8 4 16 0x y x y
2 2( 8 ) ( 4 ) 16x x y y 2 2( 8 ) ( 4 ) 1616 4 16 4x x y y
2 2( 4) ( 2) 4x y
2 2 2( 4) ( 2) 2x y
h=-4 k=-2 r=2
We use the center,
( , ) ( 4, 2),
and the radius, 2,
to graph the circle.
h k
r
2 2
The graph of
(x+4) ( 2) 4y
Practice Exercise
2 2
Complete the square and write the
equation 4 12 9 0
in standard form. Then give the
center and radius of the circle
and graph the equation.
x y x y
Answer
2 2( 2) ( 6) 49
Center (2,6)
Radius 7
x y