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Conic Concepts -- Ellipses

Module by: Kenny M. Felder. E-mail the author

Summary: This module introduces the definition and formula of an ellipse, including example problems.

The Definition of an Ellipse

An ellipse is a sort of squashed circle, sometimes referred to as an oval.

Definition of an Ellipse

Take two points. (Each one is a focus; together, they are the foci.) An ellipse is the set of all points in a plane that have the following property: the distance from the point to one focus, plus the distance from the point to the other focus, is some constant.

They just keep getting more obscure, don’t they? Fortunately, there is an experiment you can do, similar to the circle experiment, to show why this definition leads to an elliptical shape.

Experiment: Drawing the Perfect Ellipse

  1. Lay a piece of cardboard on the floor.
  2. Thumbtack one end of a string to the cardboard.
  3. Thumbtack the other end of the string, elsewhere on the cardboard. The string should not be pulled taut: it should have some slack.
  4. With your pen, pull the middle of the string back until it is taut.
  5. Pull the pen all the way around the two thumbtacks, keeping the string taut at all times.
  6. The pen will touch every point on the cardboard such that the distance to one thumbtack, plus the distance to the other thumbtack, is exactly one string length. And the resulting shape will be an ellipse. The cardboard is the “plane” in our definition, the thumbtacks are the “foci,” and the string length is the “constant distance.”
Figure 1
A picture demonstrating how to draw a perfect ellipse with thumbtacks and string.

Do ellipses come up in real life? You’d be surprised how often. Here is my favorite example. For a long time, the orbits of the planets were assumed to be circles. However, this is incorrect: the orbit of a planet is actually in the shape of an ellipse. The sun is at one focus of the ellipse (not at the center). Similarly, the moon travels in an ellipse, with the Earth at one focus.

The Formula of an Ellipse

With ellipses, it is crucial to start by distinguishing horizontal from vertical.

Table 1: Mathematical Formula for an Ellipse with its Center at the Origin
Horizontal Vertical
x2a2+y2b2=1x2a2+y2b2=1 size 12{ { {x rSup { size 8{2} } } over {a rSup { size 8{2} } } } + { {y rSup { size 8{2} } } over {b rSup { size 8{2} } } } =1} {} (a>b) x2b2+y2a2=1x2b2+y2a2=1 size 12{ { {x rSup { size 8{2} } } over {b rSup { size 8{2} } } } + { {y rSup { size 8{2} } } over {a rSup { size 8{2} } } } =1} {} (a>b)
Picture of an horizontal ellipse with parts labeled Picture of an vertical ellipse with parts labeled

And of course, the usual rules of permutations apply. For instance, if we replace xx with x-hx-h, the ellipse moves to the right by hh. So we have the more general form:

Table 2: Mathematical Formula for an Ellipse with its Center at xxx(h,k)
Horizontal Vertical
(xh)2a2+(yk)2b2=1(xh)2a2+(yk)2b2=1 size 12{ { { \( x - h \) rSup { size 8{2} } } over {a rSup { size 8{2} } } } + { { \( y - k \) rSup { size 8{2} } } over {b rSup { size 8{2} } } } =1} {} (a>b)(a>b) (xh)2b2+(yk)2a2=1(xh)2b2+(yk)2a2=1 size 12{ { { \( x - h \) rSup { size 8{2} } } over {b rSup { size 8{2} } } } + { { \( y - k \) rSup { size 8{2} } } over {a rSup { size 8{2} } } } =1} {} (a>b)(a>b)

The key to understanding ellipses is understanding the three constants aa, bb, and cc.

Table 3
  Horizontal Ellipse Vertical Ellipse
Where are the foci? Horizontally around the center Vertically around the center
How far are the foci from the center? c c c c
What is the “major axis”? The long (horizontal) way across The long (vertical) way across
How long is the major axis? 2 a 2a 2 a 2a
What is the “minor axis?” The short (vertical) way across The short (horizontal) way across
How long is the minor axis? 2 b 2b 2 b 2b
Which is biggest? aa is biggest. a > b a>b, and a > c a>c. a a is biggest. a > b a>b, and a > c a>c.
crucial relationship a 2 = b 2 + c 2 a 2 = b 2 + c 2 a 2 = b 2 + c 2 a 2 = b 2 + c 2

The following example demonstrates how all of these concepts come together in graphing an ellipse.

Example 1: Graphing an Ellipse

Table 4
Graph x 2 + 9 y 2 4 x + 54 y + 49 = 0 x 2 +9 y 2 4x+54y+49=0 The problem. We recognize this as an ellipse because it has an x 2 x 2 and a y 2 y 2 term, and they both have the same sign (both positive in this case) but different coefficients (3 and 2 in this case).
x 2 4 x + 9 y 2 + 54 y = -49 x 2 4x+9 y 2 +54y=-49 Group together the xx terms and the yy terms, with the number on the other side.
( x 2 4 x ) + 9 ( y 2 + 6 y ) = -49 ( x 2 4x)+9( y 2 +6y)=-49 Factor out the coefficients of the squared terms. In this case, there is no x 2 x 2 coefficient, so we just have to factor out the 9 from the y y terms.
( x 2 4 x + 4 ) + 9 ( y 2 + 6 y + 9 ) = -49 + 4 + 81 ( x 2 4x+4)+9( y 2 +6y+9)=-49+4+81 Complete the square twice. Remember, adding 9 inside those parentheses is equivalent to adding 81 to the left side of the equation, so we must add 81 to the right side of the equation!
( x - 2 ) 2 + 9 ( y + 3 ) 2 = 36 (x-2 ) 2 +9(y+3 ) 2 =36 Rewrite and simplify. Note, however, that we are still not in the standard form for an ellipse!
( x - 2 ) 2 36 + ( y + 3 ) 2 4 = 1 ( x - 2 ) 2 36 + ( y + 3 ) 2 4 =1 Divide by 36. This is because we need a 1 on the right, to be in our standard form!
Center: (2,–3) We read the center from the ellipse the same way as from a circle.
a= 6a=6 b = 2 b=2 Since the denominators of the fractions are 36 and 4, a a and b b are 6 and 2. But which is which? The key is that, for ellipses, aa is always greater than bb. The larger number is aa and the smaller is bb.
Horizontal ellipse Going back to the equation, we see that the a 2 a 2 (the larger denominator) was under the xx, and the b2b2 (the smaller) was under the yy. This means our equation is a horizontal ellipse. (In a vertical ellipse, the a 2 a 2 would be under the y y.)
c = 32 = 4 22 c= 32 =4 22 (approximately 5 1 2 51 2) We need cc if we are going to graph the foci. How do we find it? From the relationship a 2 = b 2 + c 2 a 2 = b 2 + c 2 which always holds for ellipses.
A horizontal ellipse centered at (2,-3). So now we can draw it. Notice a few features: The major axis is horizontal since this is a horizontal ellipse. It starts aa to the left of center, and ends aa to the right of center. So its length is 2 a 2a, or 12 in this case.The minor axis starts b b above the center and ends b b below, so its length is 4.The foci are about 5 1 2 5 1 2 from the center.

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