Understanding the Length of Axes in an Ellipse Equation

[kripkrip420]

Homework Statement

As I was relearning some concepts in calculus, I came across a section on ellipses. What I don't understand is why a and b in the standard equation of an ellipse govern the length of the minor/major axes. Can anyone shed some light? Thank you very much!

Homework Equations

The Attempt at a Solution

I have not attempted a solution, only tried to visualize the results in my head.

 

[Zryn]

Can you show us which equation you mean when you say 'Standard Equation', as there are several different ways which the equation of the ellipse can be presented. What you refer to as 'a and b' needs some context.

Either way, the co-efficient of x (when y = 0) dictates the x-intercepts (this distance between the two x points represents one axis) and the co-efficient of y (when x = 0) dictates the y-intercepts (this distance between the two y points represents the other axis). You get two points in each instance of course because to solve for x or y you have to take a square root on both sides of the equation, and thus get a +/- number.

A fairly common ellipse is 9x^2 + 4y^2 = 36. Solve for x (when y = 0) and then solve for y (when x = 0) and then plot the 4 points and join them to see the ellipse and its major and minor axes.

The special case of these distances being equal occurs when the ellipse is a circle.

Other equations involve x^2 and y^2 being fractions and always equaling 1. For example x^2/4 + y^2/9 = 1 is the same as 9x^2 + 4y^2 = 36 if you rearrange things.

 

[kripkrip420]

Thank you for the quick reply! The equation I was referring to was (x^2/a^2)+(y^2/b^2)=1.

 

[HallsofIvy]

That is the equation of an ellipse with center at (0, 0) and with its axes of symmetry along the x and y axes.

Look at what happens if x or y is 0.

If x= 0, then $0^2/a^2+ y^2/b^2= y^2/b^2= 1$ so $y^2= b^2$ and $y= \pm b$.

If y= 0, then $x^2/a^2+ 0^2/b^2= x^2/a^2= 1$ so $x^2= a^2$ and $x= \pm a$.

On the other hand, if x is not 0, then, since a square is never negative, $y^2/b^2$ must be less than 1 so y must be between -b and b. If y is not 0 then $x^2/a^2$ must be less than 1 so x must be between -a and a. That is, the ellipse goes form (-a, 0) to (a, 0) on the x-axis and from (0, -b) to (b, 0) on the y-axis.