2. Lecture Objectives
Upon completion of this chapter, you should be
able to:
Learn basic concepts about cartesian plane
Plot a coordinate in the cartesian plane
Prove geometric theorems analytically
2
3. Outline
Introduction
Rectangular coordinate system
Variable and functions
Graph of an equations
Intersection of graphs
Directed line segment
Distance between two lines
Mid point formula
3
8. Example
Draw the triangles
whose vertices
are
(a) (2,-l), (0,4),
(5,1);
(b) (2, -3),(4,4),
(-2,3).
8
9. Variable and functions
DEF: If a definite value or set of values of a
variable y is determined when a variable x takes
any one of its values, then y is said to be a
function of x.
9
Independent
variable
Independent
variable
Dependent
variable
Dependent
variable
11. Graph of an equation
DEF: The graph of an equation consists of all
the points whose coordinates satisfy the
given equation.
Techniques in graphing
Intercepts:
x intercept, let y=0
y intercept, let x=0
Assign values to the independent variable
11
13. Intersections of graphs.
If the graphs of two equations in two
variables have a point in common, then, from
the definition of a graph, the coordinates of
the point satisfy each equation separately.
Equation 1 = Equation 2
13
14. Directed lines and segments.
DEF: A line on which one direction is defined
as positive and the opposite direction as
negative is called a directed line.
14
“the shortest
distance between
two points is a line”
17. Examples
1. Find the distance between P(-3,1)
and Q(2,4).
( ) ( )( ) ( )
2 2
d P,Q 2 3 4 1
25 9
34
= − − + −
= +
=
17
18. Examples
2. 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
= − − + −
= + −
= −
− = ±
= = −
18
20. Examples
1. Find the midpoint of the line
segment whose endpoints are P(2,4)
and Q(6,3).
PQ
2 6 4 3
M ,
2 2
7
4,
2
+ +
= ÷
= ÷
20
21. Inclination and slope of a line.
The inclination of a slant line is a positive
angle less than 180 degrees
The slope of a line is defined as the tangent
of its angle of inclination.
A line which leans to the right has a positive slope
The slopes of lines which lean to the left are
negative.
The slope of a horizontal line is zero.
Vertical lines do not have a slope,
21
23. Slope
The slope m of a line passing through two
given points P1(x1,y1) and P2(x2,y2) is equal
to the difference of the ordinates divided by
the difference of the abscissas taken in the
same order; that is
23
24. Given the points A(-2,-l),B (4,0), C(3,3), and
D(-3,2), show that ABCD is a parallelogram.
24
Solution:
Examples
25. Examples
1. Find the slope of the line passing
through the points P(3,-2) and
Q(1,4) .
( )4 2 6
m 3
1 3 2
− −
= = = −
− −
25
26. Examples
If the slope of the line joining B(4, 3)
and C(b, 2) is 6, find the value of b.
( )
2 3
6
b 4
6 b 4 2 3
6b 24 1
6b 23
23
b
6
−
=
−
− = −
− = −
=
= 26
27. Parallel lines
Two non vertical lines are parallel if and only
if their slopes are equal.
27
28. Perpendicular line
Two slant lines are perpendicular if, and only
if, the slope of one is the negative reciprocal
of the slope of the other.
28
29. Angle between two lines.
Two intersecting lines form four angles.
29
31. 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 3
center : M , ,
2 2 2 2
2 5 1 0 49 1 5 2
radius : r
2 2 2
+ − + −
= = ÷ ÷
− − + − +
= = =
31
32. Analytic
Geometry
Lecture 1:Equation of the
Line
Engr. Adriano Mercedes H. Cano Jr.
University of Mindanao
College of Engineering Education
Electronics Engineering
MATH 201
32
33. THE STRAIGHT LINE
The straight line is the simplest geometric
curve.
the graph of a first degree equation in x and y
is a straight line
The locus of a first degree equation.
33
where A, B and C are constants and not
both A and B are zero.
34. Example
Determine 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.
34
35. Various Forms of an Equation
of a Line.
Slope-Intercept FormSlope-Intercept Form slope of the line
intercept
y mx b
m
b y
= +
=
= −
36. Various Forms of an Equation
of a Line.
Standard FormStandard Form , , and are integers
0, must be postive
Ax By C
A B C
A A
+ =
>
37. Various Forms of an Equation
of a Line.
Point-Slope FormPoint-Slope Form
( )
( )
1 1
1 1
slope of the line
, is any point
y y m x x
m
x y
− = +
=
38. Various Forms of an Equation
of a Line.
Intercept FormIntercept Form
39. Example
Find the intercept form and the general
equation of the line passing through the
points (2,0) and (0,1).
x y
1
2 1
x 2y 2
x 2y 2 0
+ =
+ =
+ − =
39
40. Various Forms of an Equation
of a Line.
Two point formTwo point form
41. Example
Find the general equation of the line
passing through the points (3, 2) and
(-2,-1).
( )
( )
1 2
y 2 x 3
2 3
3
y 2 x 3
5
5y 10 3x 9
3x 5y 1 0
− −
− = −
− −
− = −
− = −
− + =
41
42. Example
Find 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
− = − −
= − +
+ − =
42
48. 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
