Understanding the Length of Axes in an Ellipse Equation

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In summary, the standard equation of an ellipse, (x^2/a^2)+(y^2/b^2)=1, shows that the values of a and b represent the length of the minor and major axes, respectively. By solving for x and y, it can be seen that the x-intercepts are determined by the coefficient of x and the y-intercepts are determined by the coefficient of y. The special case of a and b being equal results in a circle.
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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!


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The Attempt at a Solution


I have not attempted a solution, only tried to visualize the results in my head.
 
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  • #2
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.
 
  • #3
Thank you for the quick reply! The equation I was referring to was (x^2/a^2)+(y^2/b^2)=1.
 
  • #4
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 [itex]0^2/a^2+ y^2/b^2= y^2/b^2= 1[/itex] so [itex]y^2= b^2[/itex] and [itex]y= \pm b[/itex].

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

On the other hand, if x is not 0, then, since a square is never negative, [itex]y^2/b^2[/itex] must be less than 1 so y must be between -b and b. If y is not 0 then [itex]x^2/a^2[/itex] 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.
 

What is the equation of an ellipse?

The general equation of an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes respectively.

What do the variables in the equation represent?

The variables (h,k) represent the coordinates of the center of the ellipse, while a and b represent the lengths of the semi-major and semi-minor axes respectively. The semi-major axis is the longest distance from the center to the edge of the ellipse, while the semi-minor axis is the shortest distance.

Can an ellipse have a negative value for a or b?

No, the values of a and b in the equation must be positive. If the value of a is larger than b, then the ellipse is longer horizontally and is called an oblong or horizontal ellipse. If the value of b is larger than a, then the ellipse is longer vertically and is called a vertical ellipse.

How does the eccentricity affect the shape of an ellipse?

The eccentricity, represented by the letter e, is a measure of how elongated or circular an ellipse is. The value of e ranges from 0 to 1, with 0 representing a perfect circle and 1 representing a straight line. The closer e is to 1, the more elongated the ellipse becomes.

What are some real-world applications of the equation of an ellipse?

The equation of an ellipse is used in many fields such as astronomy, engineering, and physics. Some examples of real-world applications include predicting the orbits of planets and satellites, designing curved bridges and arches, and creating accurate models of planetary motion.

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