m36 1.1 analytic geometry
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Leithold, L., The Calculus, 7th Edition
Leithold, L., The Calculus with Analytic Geometry
Stewart, J., Calculus: Early Transcendentals
Cuaresma, G. A., et al., A Worktext in Analytic Geometry and Calculus 1
References
Some important house rules
1. A student whose absences in both lecture and recitation exceed 15 hours may be given a grade of 5.0.
2. Attendance will be checked during the first 15 minutes of the period.
3. No one is allowed to eat, drink or smoke inside the lecture hall.
4. Avoid tardiness. If you came in late (8:40 onwards), you have to sing a non-offending song (chorus only) in front of the class otherwise you will be marked absent for that day.
5. Refrain from using cellphones, game and audio-video devices, laptops and other gadgets during the class.
Some important house rules
CHAPTER 1Analytic Geometry and
Conic SectionsMathematics Division
Institute of Mathematical Sciences and PhysicsCollege of Arts and Sciences
U. P. Los Baños
MATH 36
4
Chapter Objectives
Upon completion of this chapter, you should be able to:
find equation of a line prove geometric theorems analytically identify and sketch the graphs of second-
degree equations
5
1.1 Analytic Geometry and Analytic Proof
Objectives: Upon completion of this section, you should be able to:
recall the following concepts: distance midpoint slope
determine the general form and the other forms of equation of a line;
6
1.1 Analytic Geometry and Analytic Proof
Objectives: Upon completion of this section, you should be able to:
use Cartesian coordinate system to label parts of some known geometric figures;
use Cartesian coordinate system to prove some statements in geometry.
7
Review
8
Distance Formula
DEF: The distance between two points P(x1, y1) and Q(x2, y2) on the same plane is given by the formula
( ) ( ) ( ) 212
212 yyxxQ,Pd −+−=
)y,x(P 11
)y,x(Q 22
)y,x(P 11
1x 2x
1y
2y
(Note: )PQ( ) =Q,Pd9
Examples1. Find the distance between P(-3,1)
and Q(2,4).
( ) ( )( ) ( )2 2d P,Q 2 3 4 1
25 9
34
= − − + −
= +
=
10
Examples2. If the distance between P(-2,4) and
Q(1,y) is 5, find the value(s) of y.
( )( ) ( )( )
( )
2 2
2
2
5 1 2 y 4
25 9 y 4
16 y 4
y 4 4
y 0 or y 8
= − − + −
= + −
= −− = ±= = −
11
,2xx
x 21 +=
Midpoint FormulaDEF: The midpoint M of the line segment joining the points P(x1, y1) and Q(x2, y2) has coordinates M(x, y) where,
2yy
y 21 +=
x
y
1x 2x
1y
2y
P
M
Q
x
12
Examples1. Find the midpoint of the line segment whose endpoints are P(2,4) and Q(6,3).
PQ
2 6 4 3M ,
2 2
74,
2
+ + = ÷ = ÷
13
2. If (2, 1) and (-5, 0) are endpoints of a diameter of a circle, find the
center and radius of the circle.
Examples
( )
( )( ) ( )2 2
2 5 1 0 3 3center : M , ,
2 2 2 2
2 5 1 0 49 1 5 2radius : r
2 2 2
+ − + − = = ÷ ÷
− − + − += = =
14
Slope of a Line
DEF: The slope m of the line joining two points P(x1, y1) and Q(x2, y2) is given by
12
12
xx
yym
−−
=P
Q
12 yy −
12 xx −
(Note: The slope of a vertical line is undefined.)
15
Examples1. Find the slope of the line passing
through the points P(3,-2) and Q(1,4) .
( )4 2 6m 3
1 3 2
− −= = = −
− −
16
Examples
2. If the slope of the line joining B(4, 3) and C(b, 2) is 6, find the value of b.
( )
2 36
b 46 b 4 2 3
6b 24 1
6b 23
23b
6
−=−
− = −− = −=
= 17
Parallel lines and Perpendicular Lines
THM: Two non-vertical lines and are said to be parallel if and only if their slopes are equal.
L1L2
18
Parallel lines and Perpendicular Lines
THM: Two non-vertical and non-horizontal lines and are said to be perpendicular if and only if the product of their slopes is -1.
L1L2
19
ExampleThe slope of line L1 is 2. Determine the slope of a second line L2 if L1 and L2 are a. parallel b. perpendicular
a. If L1 and L2 are parallel then their slopes are equal. Thus, the slope of L2 is also 2.
b. If L1 and L2 are perpendicular then the product of their slopes is equal to -1. Thus, the slope of L2 is -1/2. 20
The General Form and Other Forms of Equation of a Line
21
General Equation of a Line
DEF: The general equation of a line is given by
where A, B and C are constants and not both A and B are zero.
0=++ CByAx
22
ExampleDetermine if the given points lie on the line given by .02y4x =+−
( )0,1a. ( )1,2b.
a. (1, 0) is not on the given line.
b. (2, 1) lies on the given line.
23
Two-point Form Equation of the Line
Let P(x1, y1) and Q(x2, y2) be known points on a line then the two-point form equation of a line or equation of a line in two-point form is given by
( )112
121 xx
xxyy
yy −−−=−
24
ExampleFind the general equation of the line passing through the points (3, 2) and (-2,-1).
( )
( )
1 2y 2 x 3
2 33
y 2 x 35
5y 10 3x 9
3x 5y 1 0
− −− = −− −
− = −
− = −− + =
25
Intercept Form Equation of a Line
Suppose a and b are the x-intercept and y-intercept, respectively, of a line such that a and b are nonzeros. Then the intercept form equation of a line or equation of a line in intercept form is given by
1=+b
y
a
x
26
ExampleFind the intercept form and the general equation of the line passing through the points (2,0) and (0,1).
x y1
2 1x 2y 2
x 2y 2 0
+ =
+ =+ − =
27
Point-slope Form Equation of a Line
Let m be the slope of a line that passes through the point (x1,y1) then the point-slope form equation of a line or equation of a line in point-slope form is given by
)( 11 xxmyy −=−
28
ExampleFind the general equation of the line given a slope equal to -1 and x-intercept equal to 6.
( )
( )
6 , 0 is on the line
y 0 1 x 6
y x 6
x y 6 0
− = − −= − ++ − =
29
Slope-intercept Form Equation of a Line
Let m is slope of the line and b be the y-intercept of that line then the slope-intercept form equation of a line or equation of a line in slope-intercept form is given by
bmxy +=
30
ExampleFind the general equation of line L passing through the point (-7,-5) and perpendicular to the line given by
019y4x3 =−+
3x 4 y 19 0
4 y 3x 19
3 19y x
4 4
+ − == − +−= +
( ) ( )( )
L
4m
34
y 5 x 73
???
= −
− − = − − −
31
Distance Between a Point and a Line
The distance d from a point (x1, y1) to a line L with equation Ax + By + C = 0 is given by
2211
BA
CByAxd
+++
=
End32
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