Cal 3

Calculus And Analytical

Geometry

MTH-310

Lecture 3

In this lecture we shall

Distance

Midpoint Formula

Slope

Lines

Symmetries of Graph

Equation of Circle

Distance

The distance between to point and is 1p 2p

2 2

1 2 2 1 2 1,d p p a a b b

Midpoint Formula

The midpoint M of segment is 1 2p p

1 2 1 2 1 2

1 1,

2 2M p p x x y y

Example

Given A(-2,3) and B(4,-2), find:

(a)The distance between A and B

(b) The midpoint M of segment AB

Slope

The ratio of vertical change (rise) to horizontal change

(run) of a line.

or

Slope basically describes the steepness of a line

12

12

xx

yym

Example:

In each part find the slope of the line through

If a line goes up from left to right, then the slope has to be positive

If a line goes down from left to right, then the slope has to be negative

Positive Negative

Horizontal lines have a slope of zero while vertical lines have no slope

Horizontal Vertical m = 0

m = no

slope

INTERCEPTS

An intercept of a line is a point where a line crosses an axis.

The y-intercept is the point at which the line crosses the y-axis.

The x-intercept is the point at which the line crosses the x-axis.

Equation of a Straight Line

Point – Slope Form

Slope-Intercept Form

Point – Slope Form

Given a point and the slope of a line we can

write the equation of the line in

point – slope form

Given a line passes through point (-3,5) and has a slope of -¾.Write

an equation of the line.

Write the equation of the line that goes through the points (6, –4) and

(2, 8) .

Slope-Intercept Form

Slope – Intercept Form:

y = mx + b

m = the slope of the line … b = the y-intercept

EXAMPLE: WHAT IS AN EQUATION OF THE LINE

1

5m 0, 3 and the y-intercept is

Example: y = 3x – 6

Find the equation of lines

0 5 4 3 2 1

1

2

3

4

5

-1

-4

-5

-5

-3

-4

-2

-3

-1

-2 x

y

A

B

C

D

E

F

J

K

Parallel Lines

Two lines with the same slope are said to be parallel lines. If you

graph them they will never intersect.

We can decide algebraically if two lines are parallel by finding the

slope of each line and seeing if the slopes are equal to each other.

Testing if Lines are Parallel

Are the lines parallel? 12 3 9 and -8 2 14x y x y

Practice Constructing Parallel Lines

Find the equation of the line going through the point (4,1) and

parallel to 3 7y x

Find the equation of the line going through the point (-2,7) and

parallel to 2 8x y

Perpendicular Lines

Perpendicular lines are lines that intersect in a right angle.

We can decide algebraically if two lines are perpendicular by finding the

slope of each line and seeing if the slopes are negative reciprocals of

each other. This is equivalent to multiplying the two slopes together and

seeing if their product is –1.

Testing if Lines Are Perpendicular

1Are the lines 2 5 and 4 perpendicular?

2x y y x

Constructing Perpendicular Lines

Find the equation of a line going through the point (3, -5) and

perpendicular to 2 83

y x

Find the equation of the line going through the point (4,1) and

perpendicular to 3 7y x

Quadratic Equation

ax2 + bx + c = 0, where a ≠ 0.

Methods Used to Solve Quadratic Equations

By Factorization

By Completing Square Method

Quadratic Formula

Symmetry of Graph

There are three type of symmetry

(1)Symmetry about x-axis

(2)Symmetry about y-axis

(3)Symmetry about origin

2

2

3

1( )

2

( )

( )4

a y x

b y x

c y x

Equation of a Circle

The center of a circle is given by (h, k)

The radius of a circle is given by r

The equation of a circle in standard form is

(x – h)2 + (y – k)2 = r2

Example

Find an equation of circle that has center C(-2,3) and

contains the point D(4,5).

Identify the center and radius and sketch the graph:

253422

yx

Identify the center and radius and sketch the graph:

6499 22 yx