# Exploring the Equation of an Ellipse

Lesson Plans > Mathematics > Geometry > Analytic Geometry > Conic Sections > Ellipses## Exploring the Equation of an Ellipse

How does the equation of an ellipse relate to its graph?

I Introduce the ellipse concept

"An ellipse looks like an oval. Pretty much a circle that is wider or taller in 1 of its dimensions. You can see pictures

of ellipses here http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php"

II. Exploring the equation and its graph

(To do the following part of the exercise , please open up www.meta-calculator.com, an online graphing calculator that can graph

ellipses. We will explore the way that changing the equation affects the graph of the ellipse)

The general equation of an ellipse looks like x^{2}/a +y^{2}/b = 1 .

Now lets explore the roles of each of these 2 variables. First, let's make b =5 and try exploring different values for 'a'

Graph the following equations, making note of the graph of each:

x^{2}/1 +y^{2}/5 = 1 , a = 1

x^{2}/2 +y^{2}/5 = 1 , a = 2

x^{2}/3 +y^{2}/5 = 1 , a = 3

x^{2}/4 +y^{2}/5 = 1 , a = 4

x^{2}/5 +y^{2}/5 = 1 , a = 5

What effect does increasing 'a'have on the equation ?

Now, take a guess, what do you think will be the effect of changing the variable underneath the y^{2} term? For instance, if the

term under y^{2} starts at 1 and then increases, predict how the graph will change:

x^{2}/5 +y^{2}/1 = 1 , b = 1

x^{2}/5 +y^{2}/2 = 1 , b = 2

x^{2}/5 +y^{2}/3 = 1 , b = 3

x^{2}/5 +y^{2}/4 = 1 , b = 4

x^{2}/5 +y^{2}/5 = 1 , b = 5

Ok, was your prediction correct?

Concluding question, in terms of 'a' and 'b', what is true when the ellipse becomes a circle?