10 - 3   T h e  E l l i p s e

by Elizabeth R

~ An ellipse is somewhat like a circle, but with uneven lengths of its' axis.  It has two axes, a major and minor, two foci points, a center or origin.  There are two standard equations for an ellipse based on which axis is the major.  The two equations are as follows:            (x-h)² + (y-k)² = 1     (y-k)² + (x-h)² = 1 1                          a²           b²                        a²           b²

^{~To solve an ellipse, you must find a, b, h, and k based on the standard equations.  These will indicate where to place key points when graphing an ellipse.  To find the center: you look at the values of h and k, switch the sign, and place them in (x,y) order.  To find all of these key points, the equation must be equal to one and in standard form.  We can recognize that the conic equation is an ellipse if we see that there are two squared terms with different coefficients when the equation is equal to 1.}  

~First, we will start simple and learn how to put the equation in standard form.  If we are given the problem x²+4y²-8y+6x+9=0.   (x²+6x+___) + 4(y²-2y+___)=-9 + __+__  This is completing the square.  (x²+6x+9) + 4(y²-2y+1)=-9+9+4 Next, we put it into standard form (x+3)² +4(y-1)²=4  Because the equation has to be equal to 1, we divide everything by 4.  (x+3)²/4 + (y-1)²/1=1.

~Next, we are going to use this equation to find the center.  The center for this problem will be at (-3, 1) because you switch the sign that corresponds to the x value and the y value.  

~After we find the center, we need to locate the points of the foci and the vertices.  To find the foci, take the square root of (big-small).  This means look at the denominators and see which one is larger.  After doing this, take the square root of the big one minus the smaller one.  This will give you a number that we will use later.  For this problem, the foci value will be square root of(4-1) because the bigger valued denominator is 4 while the smaller value is 1.  This value ends up being the square root of 3.

~These denominators are helpful in determining the vertices also.  The denominator for the x term tells us how far to go left and right from the center while the denominator for the y term tells us how far to go up and down from the center.  For any problem, you take the value of a and b and then add and subtract them vertically if they correspond to the y and horizontally if they correspond to the x.  In this problem, we will take the square root of 4, or 2, and add it and subtract it from the center (-3,1).  Because we are moving horizontally here, we keep the y value constant and add/subtract 2 to x.  Two of our vertices are (-3+2,1) and (-3-2,1) or(-1,1),(-5,1) We find the other two the same way.  We take the square root of 1 or 1 and then add/subtract it from the center but this time we will keep x constant and add/subtract from the y.  The two other vertices are (-3, 1+1) and (-3, 1-1).  

~We have almost finished the long process of explanation.  We are going to revisit the foci point and figure out the points because foci are points on a graph, not just a number.  Because x is listed first in our standard equation, we have to add and subtract from the center x value to get the foci points.  Remember that our number we got to use in the foci is the square root of 3.  So, the values of the foci will be (-3+square root of 3,1) and (-3-square root of 3,1).  

WE HAVE FINISHED FINDING KEY POINTS, NOW WE GET TO DO THE FUN PART!! WE HAVE TO GRAPH THE ELLIPSE.  To begin graphing, plot the center and vertex points.  Then plot the foci and label them F one and F two.  An ellipse is an oval so make it look like one.  The graph will look like this:           

The green dot is the center, blue are foci and the pink is the actual ellipse.  Although it is hard to see, this is the ellipse.  It is very easy to draw.  

Now it is time for a word problem.  Suzie notices that the center of her dog pin is at (0,0) in her coordinate back yard.  She wants to write the standard form of the equation of this ellipse-shaped dog pin after she realizes that the dog pin goes 4 units out from the center both ways and three units up both ways from the center.  Help her write the equation and then find the foci.  

Answer:  The equation would be (x-0)² + (y-0)² =1                         1                                                        16           9   

This is because 4 represents a and 3 represents b so we plug these numbers in as denominators.  The center is (0,0) so there are no numbers subtracted or added in the numerators.  Good Job, you finished the word problem!

No webpage is good without a little inspiration so here is a quote for you:  

        This quote means that you should never hold back your creativity because that is deprivation and you need to always give all your mind or you will deprive yourself of what you can really accomplish.  This has to do with the character trait perseverance because it has to do with giving your all all the time and never being frugal.  

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