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Ellipses: From Definition to Equation

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

Summary: A teacher's guide to the connection between the definition and equation of an ellipse, and how to get from one to the other.

Table 1
A picture of an ellipse showing the relationship between an ellipse and its foci. Here is the geometric definition of an ellipse. There are two points called the “foci”: in this case, (-3,0) and (3,0) . A point is on the ellipse if the sum of its distances to both foci is a certain constant: in this case, I’ll use 10 . Note that the foci define the ellipse, but are not part of it.

The point ( x x , y y) represents any point on the ellipse. d 1 d1 is its distance from the first focus, and d 2 d2 to the second. So the ellipse is defined geometrically by the relationship: d 1 + d 2 = 10 d1+d2=10.

To calculate d 1 d1 and d 2 d2, we use the Pythagorean Theorem as always: drop a straight line down from ( x x, y y) to create the right triangles. Please verify this result for yourself! You should find that d 1 = (x+3)2+y2d1=(x+3)2+y2 size 12{ sqrt { \( x+3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {} and d 2 = (x3)2+y2d2=(x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}. So the equation becomes:

(x+3)2+y2+ (x3)2+y2= 10 (x+3)2+y2 size 12{ sqrt { \( x+3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}+(x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}=10. This defines our ellipse

The goal now is to simplify it. We did problems like this earlier in the year (radical equations, the “harder” variety that have two radicals). The way you do it is by isolating the square root, and then squaring both sides. In this case, there are two square roots, so we will need to go through that process twice.

Table 2
(x+3)2+y2= 10 (x3)2+y2(x+3)2+y2 size 12{ sqrt { \( x+3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}=10(x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {} Isolate a radical
( x + 3 ) 2 + y 2 = 100 20 (x3)2+y2+ ( x 3 ) 2 + y 2 (x+3 ) 2 + y 2 =10020(x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}+(x3 ) 2 + y 2 Square both sides
( x 2 + 6 x + 9 ) + y 2 = 100 20 (x3)2+y2+ ( x 2 6 x + 9 ) + y 2 ( x 2 +6x+9)+ y 2 =10020(x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}+( x 2 6x+9)+ y 2 Multiply out the squares
12 x = 100 20 (x3)2+y212x=10020(x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {} Cancel & combine like terms
(x3)2+y2= 5 35x (x3)2+y2 size 12{ sqrt { \( x - 3 \) rSup { size 8{2} } +y rSup { size 8{2} } } } {}=535 size 12{ { {3} over {5} } } {}x Rearrange, divide by 20
( x 3 ) 2 + y 2 = 25 6 x + 925 x 2 (x3 ) 2 + y 2 =256x+925 size 12{ { {9} over {"25"} } } {} x 2 Square both sides again
( x 2 6 x + 9 ) + y 2 = 25 6 x + 925x 2 ( x 2 6x+9)+ y 2 =256x+925 size 12{ { {9} over {"25"} } } {}x 2 Multiply out the square
1625 x 2 + y 2 = 16 1625 size 12{ { {"16"} over {"25"} } } {} x 2 + y 2 =16 Combine like terms
x 2 25 + y 2 16 = 1 x 2 25 + y 2 16 = 1 size 12{ { {x rSup { size 8{2} } } over {"25"} } + { {y rSup { size 8{2} } } over {"16"} } =1} {} Divide by 16

…and we’re done! Now, according to the “machinery” of ellipses, what should that equation look like? Horizontal or vertical? Where should the center be? What are a a, b b, and c c? Does all that match the picture we started with?

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