2.2 parallel and perpendicular lines and circles slopes and parallel lines 1. if two nonvertical...

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