# Ellipse

Math > Math Concepts  > Algebra > Conic Sections > Ellipse

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## Ellipse

In the following figure the plane is slicing the cone at an angle, β, greater than 900.

As β increases the plane will eventually become parallel with the edge of the cone.  At this point the section will be a parabola;  up to this point we have an ellipse.

The intersection will look like the following:

An ellipse has two axes, the major and the minor axes. Each half is called a semi axis and are the radii of the ellipse.

There are two focal points, always on the major axis, at the same distance from the center of the ellipse.  This distance we'll label with the lower case letter c. The significance of these focal points will be discussed later.

The length of the x radius we'll label with the letter a.  Now, imagine sliding the ends of the dark red line along both axis through points 1, 2 then 3.

The length of the y radius we'll label  with the  letter b (shown below.)  The distance from f1 to P is also a.  The triangle f1 p f2 is an isosceles triangle with a base equal to 2c and whose legs each equal a and whose altitude is b.

The distance from f1 to any point P on the ellipse to f2 is constant, equal to 2a. Of course, this is true for every ellipse.

The major axis can lie along the Y axis as shown below. All previous relationships hold.

In each ellipse above (horizontal and vertical) a, b, and c form the sides of a right triangle.  a and b interchange  rolls as the hypotenuse.  Using Pythagorean's Theorem, we have

b2 + c2 = a2           and       a2 + c2 = b2

c2 = a2 - b2     and              c2 = b2 - a2

c = (a2 - b2)½  and              c = (b2 - a2)½

We can combine these two equations using absolute value like so:

c   =  | a2 - b2 | ½

When you look at the horizontal and vertical ellipses you'll note that in the  first a is always larger than b, and in the second  b is always larger then a.  We're  interested in the difference of their squares, and that must be positive, so in both cases the absolute value of this difference will work.

The equation of an ellipse centered at the origin is:

As with the circle, we can translate the center of the ellipse to a point (h,k) and the like this:

and the  equation in this case is :

CAUTION:  most, if not all texts will take this standard equation and interchange the a and b to keep the 'a' with the major axis and the 'b' with the minor axis to describe a vertical ellipse.

Like so:

I believe that doing this only adds confusion. I chose NOT to identify a and b with these axes but, instead, with the  radii,  i.e., the x radius and the y radius respectively.  'a' is always associated with x and b is always associated with y.  This is simpler and  more consistent.  And it allows, without memorization, the understanding of how a and b relate to form a vertical and a horizontal ellipse.

 Examples: 1) Plot the ellipse: a = 12, b = 3, c = | 122 - 32 | 1/2 = |144 - 9 |1/2 c = 1351/2 = 11.6 Since this ellipse is centered at the origin, the focal points are ±11.6 2) Plot the ellipse: Notice the a and b values are interchanged. The larger one determines which axis is the major axis. In this case that axis is the Y axis. Also note that in these two examples the centers of the ellipses are the origin. The next example is not centered at the origin. a=3 and b = 12, c = | b2 - a2 |1/2 = |122 - 32| 1/2 = 11.6 Note: for ellipses centered at the origin, 1) the value under the x2 term is the intersections on the X axis, both positive and negative. 2) the value under the y2 term is the intersections on the Y axis, both positive and negative. CAUTION! 4 x2 + y2 = 12 IS NOT IN STANDARD FORM! We need a '1' on the right hand side of the equation. So we divide both sides of the equation by 12 like so: Now we have the equation in standard form and we may proceed. 3) Plot the ellipse: Here we have the center displaced by +1 from the origin along the x axis and +3 from the origin along the y direction. a > b so the major axis is parallel to the X axis. a = 5, b = 4 and c = |a2 - b2|1/2 = | 25 - 16 |1/2 = 91/2 = 3 The new axis (dotted) has its origin at (1, 3) We have all the information now to plot the ellipse as follows: 4) Write the equation for the graphs to the right.

This is another FREE ALGEBRA PRINTABLE presented to you from the Algebra section of K12math.com