alg ellipses

8/17/2019 Alg Ellipses

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Preface

Here are my online notes for my Algebra course that I teach here at Lamar University, although Ihave to admit that it’s been years since I last taught this course. At this point in my career I

mostly teach Calculus and Differential Equations.

Despite the fact that these are my “class notes” they should be accessible to anyone wanting to

learn Algebra or needing a refresher in for algebra. I’ve tried to make the notes as self contained

as possible and do not reference any book. However, they do assume that you’ve has some

exposure to the basics of algebra at some point prior to this. While there is some review of

exponents, factoring and graphing it is assumed that not a lot of review will be needed to remind

you how these topics work.

Here are a couple of warnings to my students who may be here to get a copy of what happened on

a day that you missed.

1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn

algebra I have included some material that I do not usually have time to cover in class

and because this changes from semester to semester it is not noted here. You will need to

find one of your fellow class mates to see if there is something in these notes that wasn’t

covered in class.

2. Because I want these notes to provide some more examples for you to read through, I

don’t always work the same problems in class as those given in the notes. Likewise, even

if I do work some of the problems in here I may work fewer problems in class than are

presented here.

3. Sometimes questions in class will lead down paths that are not covered here. I try to

anticipate as many of the questions as possible in writing these up, but the reality is that I

can’t anticipate all the questions. Sometimes a very good question gets asked in class

that leads to insights that I’ve not included here. You should always talk to someone who

was in class on the day you missed and compare these notes to their notes and see what

the differences are.

4. This is somewhat related to the previous three items, but is important enough to merit its

own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!!

Using these notes as a substitute for class is liable to get you in trouble. As already noted

not everything in these notes is covered in class and often material or insights not in these

notes is covered in class.


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Ellipses

In a previous section we looked at graphing circles and since circles are really special cases of

ellipses we’ve already got most of the tools under our belts to graph ellipses. All that we really

need here to get us started is then standard form of the ellipse and a little information on how to

interpret it.

Here is the standard form of an ellipse.

( ) ( )2 2

2 21

x h y k

a b

− −+ =

Note that the right side MUST be a 1 in order to be in standard form. The point is called

the center of the ellipse.

( ,h k )

To graph the ellipse all that we need are the right most, left most, top most and bottom most

points. Once we have those we can sketch in the ellipse. Here are formulas for finding these

points.

( )

( )

( )

( )

right most point : ,

left most point : ,

top most point : ,

bottom most point : ,

h a k

h a k

h k b

h k b

+

−

+

−

Note that a is the square root of the number under the x term and is the amount that we move

right and left from the center. Also, b is the square root of the number under the y term and is the

amount that we move up or down from the center.

Let’s sketch some graphs.

Example 1 Sketch the graph of each of the following ellipses.

(a)

( ) ( )2 2

2 41

9 25

x y+ −+ = [Solution]

(b) ( )

22 31

49 4

y x −+ = [Solution]

(c) [( ) ( )2 2

4 1 3 x y+ + + = 1 Solution]

Solution

(a) So, the center of this ellipse is ( and as usual be careful with signs here! Also, we haveand . So, the points are,

)2,4−3a = 5b =

( )

( )

( )

( )

right most point : 1,4

left most point : 5,4

top most point : 2,9

bottom most point : 2, 1

−

−

− −


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Here is a sketch of this ellipse.

[Return to Problems]

(b) The center for this part is and we have(0,3) 7a = and 2b = . The points we need are,

( )

( )

( )

( )

right most point : 7,3

left most point : 7,3

top most point : 0,5

bottom most point : 0,1

−

Here is the sketch of this ellipse.


(c) Now with this ellipse we’re going to have to be a little careful as it isn’t quite in standard form

yet. Here is the standard form for this ellipse.

( )( )

2

213 1

1

4

x y

++ + =


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Note that in order to get the coefficient of 4 in the numerator of the first term we will need to

have a1

4 in the denominator. Also, note that we don’t even have a fraction for the y term. This

implies that there is in fact a 1 in the denominator. We could put this in if it would be helpful to

see what is going on here.( ) ( )

2 21 3

11 1

4

x y+ ++ =

So, in this form we can see that the center is ( )1, 3− − and that1

2a = and . The points for

this ellipse are,

1b =

( )

( )

1right most point : , 3

2

3left most point : , 32

top most point : 1, 2

bottom most point : 1, 4

⎛ ⎞− −⎜ ⎟

⎝ ⎠

⎛ ⎞− −⎜ ⎟⎝ ⎠

− −

− −

Here is this ellipse.


Finally, let’s address a comment made at the start of this section. We said that circles are really

nothing more than a special case of an ellipse. To see this let’s assume that . In this case

we have,

a b=

( ) ( )2 2

2 21

x h y k

a a

− −+ =


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Note that we acknowledged that and used a in both cases. Now if we clear denominators

we get,

a b=

( ) ( )2 2 2 x h y k a− + − =

This is the standard form of a circle with center ( ),h k and radius a. So, circles really are specialcases of ellipses.

alg ellipses

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