Transcript 2.2 Parallel and Perpendicular Lines and Circles
2.2 Parallel and Perpendicular Lines and Circles 1.
2.
3.
Slopes and Parallel Lines If two nonvertical lines are parallel, then they have the same slopes.
If two distinct nonvertical lines have the same slope, then they are parallel.
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 line whose equation is
y
Express the equation in slope-
x
intercept form.
4.
Solution Notice that the line passes through the form of the line's equation, we have
x
1 2 and
y
1
y
1 7.
x
1 ) Y 1 =-7 X 1 =-2
What is the slope of the line?
Given equation
y x
4 Slope of the line is –5.
Parallel lines have the same slope.
So
m
5.
X 1 =-2, y 1 =-7, and m=-5
y
5(
x y y
5
x x
10 17 This is the slope-intercept form of the equation.
Practice Exercise Write an equation of the line passing line whose equation is 3
x
2
y
Express the equation in slope intercept form.
0.
Answer to the Practice Exercise
y
3 2
x
9 2
Slopes and Perpendicular Lines 1.
2.
3.
If two nonvertical lines are perpendicular, then the product of their slopes is –1.
If the product of the slopes of two lines is –1, then lines are perpendicular.
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 a
Given Line
Find the slope of any line that is perpendicular to the line whose equation is 3
x
2
y
0.
Solution Solve the given equation for y.
3
x
2
y
0 2
y x
6
y
3 2
x
3 Slope is –3/2.
Given line has slope –3/2.
Any line perpendicular to this line has a slope that is the negative reciprocal of 3 2 .
Thus, the slope of any perpendicular 2 line is .
3
Practice Exercise The equation of a line is given by 3
x
4
y
0. Find the slope of a line that is (a) parallel to the line; and (b) perpendicular to the line.
Answers (a) 3 4 (b) 4 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 The standard form of the equation of a c ircle with center (
h
,
k
) and radius
r
(
x
h
) 2 (
y
k
) 2
r
2 .
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
Solution
Radius
r
4
(
x
(
(
x
) 2 2) 2
y
(
2)
2
(
y
) 2 2
1)
2
r
2 4 2
16
Practice Exercises Write the standard form of the equation of the circle with the given center and radius.
1. Center (0, 0),
r r
8 3
Answers
1.
x
2
y
2
64 2. (
x
3)
2
(
y
5)
2
9
Example 4: Using the Standard Form of a Circle’s Equation to Graph the Circle Find the center and radius of the circle whose equation is (
x
1) 2 (
y
4) 2 25 and graph the equation.
Solution Radius 5
Practice Exercise Give the center and radius of the circle described by the equation (
x
4) 2 (
y
5) 2 36 and graph the equation.
Answer Radius 6
The General Form of the Equation of a Circle Th e
general form of the equation of a circle i s
x
2
y
2
Dx
Ey
0
.
Example 5: Converting the General Form of Circle’s Equation to Standard Form and Graphing the Circle Write in standard form and graph:
x
2
y
2 8
x
4
y
16 0.
Solution
x
2 (
x
2 8
x
y
2 8
x
4
y
16 0
y
2 4
y
) 16 (
x
2 8
x
16
y
2 4
y
4 ) 16 4 (
x
4) 2 (
y
2) 2 4
x
h=-4
y
k=-2 We use the center,
2 2 2 r=2 and the radius,
r
2, to graph the circle.
The graph of (x+4) 2 (
y
2) 2 4
Practice Exercise Complete the square and write the equation
x
2
y
2 4
x
12
y
0 in standard form. Then give the center and radius of the circle and graph the equation.
Answer
(
x
2)
2
(
y
6)
2