Analytic geometry lecture1

Analytic Geometry

Lecture 1:Lines

Engr. Adriano Mercedes H. Cano Jr.University of Mindanao

College of Engineering EducationElectronics Engineering

MATH 201

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Lecture ObjectivesUpon 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

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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

Slope3

Outline Slope Parallel lines Perpendicular lines Angle between lines

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Introduction Euclid

Greek mathematician Elements of Geometry Euclidian Geometry

Rene Descartes French mathematician, philosopher La Geomtrie (1637) Analytic Geometry

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Rectangular Coordinates

Cartesian plane

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1st Quadrant

4th Quadrant

2nd Quadrant

3rd Quadrant

Rectangular Coordinates

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(3,2)

abscissa

ordinate

coordinate

Example Draw the triangles

whose vertices are (a) (2,-l), (0,4),

(5,1); (b) (2, -3),(4,4), (-2,3).

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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.

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Independent variable

Independent variable

Dependent variable

Dependent variable

Useful notation for functions.

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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

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let

Examples

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

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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.

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“the shortest distance between two points is a line”

The distance between two points.

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Examples

Examples1. Find the distance between P(-3,1)

and Q(2,4).

2 2d P,Q 2 3 4 1

25 9

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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 4y 0 or y 8

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The mid-point of a line segment.

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Examples1. Find the midpoint of the line segment whose endpoints are P(2,4) and Q(6,3).

PQ

2 6 4 3M ,2 274,2

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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,

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Inclination and slope of a line

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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

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Given the points A(-2,-l),B (4,0), C(3,3), and D(-3,2), show that ABCD is a parallelogram.

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Solution:

Examples

Examples1. Find the slope of the line passing

through the points P(3,-2) and Q(1,4) .

4 2 6m 31 3 2

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ExamplesIf the slope of the line joining B(4, 3) and C(b, 2) is 6, find the value of b.

2 36b 4

6 b 4 2 3

6b 24 16b 23

23b6

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Parallel lines Two non vertical lines are parallel if and only

if their slopes are equal.

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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.

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Angle between two lines. Two intersecting lines form four angles.

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Examples

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 : r2 2 2

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Analytic Geometry

Lecture 1:Equation of the Line

Engr. Adriano Mercedes H. Cano Jr.University of Mindanao

College of Engineering EducationElectronics Engineering

MATH 201

32

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.

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where A, B and C are constants and not both A and B are zero.

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.

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Various Forms of an Equation of a Line.

Slope-Intercept Slope-Intercept FormForm

slope of the lineintercept

y mx bmb y


Standard Standard FormForm

, , and are integers0, must be postive

Ax By CA B CA A


Point-Slope Point-Slope FormForm

1 1

1 1

slope of the line, is any point

y y m x x

mx y


Intercept FormIntercept Form

ExampleFind the intercept form and the general equation of the line passing through the points (2,0) and (0,1).

x y 12 1x 2y 2x 2y 2 0

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Two point Two point form form

ExampleFind the general equation of the line passing through the points (3, 2) and (-2,-1).

1 2y 2 x 32 3

3y 2 x 35

5y 10 3x 93x 5y 1 0

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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 6x y 6 0

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The distance from a line to a point.

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ExampleFind the general equation of line L passing through the point (-7,-5) and perpendicular to the line given by

019y4x3

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The distance between two parallel lines

Example

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Family of lines through the intersection of two lines.

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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