How To Write The Equation Of An Ellipse

The Equation Of An Ellipse

Just like a circle, parabola and hyperbola, an ellipse is a cross section of a conical section.  In fact, an ellipse is very similar to a circle except that the cross section is taken at an angle to the conical section with gives the ellipse it’s elongated, oval shape.  An ellipse when centered at the origin (intersection of the x and y axes) of a coordinate system will have both x and y intercepts with respect to the positive and negative directions.  The basic difference between an ellipse and a circle can be seen here in that with respect to the x and y axes, the ellipse has different radii.


The definition of an ellipse states that with any two points (known as foci) at fixed locations within the ellipse that the summation of the distances from each point to any location along the curve of the ellipse will remain the same.


The standard equation of an ellipse takes the two following forms.


Ellipse centered at (0, 0):


(x2/ a2) + (y2/ b2) = 1


Ellipse centered at (h, k):


[(x –h)2/ a2] + [(y –k)2/ b2] = 1


In these ellipse equations, x and y are the coordinates at any point along the curve of the ellipse.  Respectively, a and b are the x and y intercepts and for the ellipse centered at the origin (0, 0), these are located at (a, 0), (-a, 0), (0, b) and (0,-b).  It should be noted that a is not equal to b.



A few examples will help in further explaining this concept.


1.) Determine the x and y coordinates of the following ellipse: 12 x2 + 27 y2 = 108.


If we rearrange this equation, we can write it in to the standard equation of an ellipse centered at the origin.


(12 x2 + 27 y2)(1/108)= (108)(1/108)

(12/108) x2 + (27/108) y2 = 1

x2/9 + y2/4 = 1

x2/32 + y2/22 = 1


So a = 3 and b = 2.  This means that the longer portion of the ellipse follows along the x-axis and the shorter along the y-axis.  Since this ellipse is centered about (0, 0), we can write the x-intercepts as (3, 0) and (-3, 0) and the y-intercepts as (0, 2) and (0, -2).



2.) Write the equation of the ellipse centered at (-3, 5) with a = 5 and b = 8.


We use the form, [(x –h)2/ a2] + [(y –k)2/ b2] = 1, and input our values to yield the following.


[(x +3)2/ 52] + [(y –5)2/ 82] = 1


[(x +3)2/ 25] + [(y –5)2/ 64] = 1 (This is our final equation.)





Bittinger, Marvin L., Ellenbogen, David, Keedy, Mervin L. (1994).  Intermediate Algebra: Concepts and Application (4th ed.).  USA: Addison-Wesley Publishing Company, Inc.