# How to Analyze an Ellipse

In this article, you'll learn how to analyze an Ellipse for its geometric elements and make a chart of your findings in Microsoft Excel.

## Steps

• Become familiar with the image to create: ### Part 1 The tutorial

1. 1
Accept an ellipse in standard formula format, i.e. ${\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1$ or ${\frac {(x-h)^{2}}{b^{2}}}+{\frac {(y-k)^{2}}{a^{2}}}=1$, where $a>b$.
2. 2
Find the following elements, which you also memorize the formulas for per the following KEY:
• The Center = $(h,k)$
• The Major Axis = $2a$
• The Minor Axis = $2b$
• The Vertices = $(h,k+a)$, $(h,k-a)$
• The Co-Vertices = $(h+b,k)$, $(h-b,k)$
• The Foci = $(h,k+c)$, $(h,k-c)$ for c of $b^{2}+c^{2}=a^{2}$
3. 3
Given the equation ${\frac {(x-5)^{2}}{64}}+{\frac {(y+2)^{2}}{100}}=1$, analyze the ellipse it represents into the above elements:
• Following the standard formula format, $(h,k)$ = $(5,{-2})$ because both numbers are subtracted in the standard format. Thus, the Center is at $(h,k)$ at $(5,{-2})$.
• $a^{2}>b^{2}$, so look to see which is larger of the two denominators and find that $100>64$, $a^{2}=100$ and $b^{2}=64$, so $a=10$ and $b=8$.
• Per the KEY, find that the Major Axis = $2a$ and thus = $2*10=20$.
• Per the KEY, find that the Minor Axis = $2b$ and thus = $2*8=16$.
• Per the KEY, find that the Vertices = $(h,k+a)$, $(h,k-a)$ and thus = $(5,{-2}+10)$, $(5,{-2}-10)$ so the Vertices = $(5,8)$ and $(5,{-12})$.
• Per the KEY, find that the Co-Vertices = $(h+b,k)$, $(h-b,k)$ and thus = $(5+8,{-2})$, $(5-8,{-2})$ so the Co-Vertices = $(13,{-2})$ and $({-3},{-2})$.
• You now should be able to sketch the graph with pencil and paper, given the above successful completion of finding the elements so far.
• Lastly per the KEY, find that the Foci = $(h,k+c)$, $(h,k-c)$ for $c$ of $b^{2}+c^{2}=a^{2}$. You know that $b^{2}=64$ and $a^{2}=100$, so $c^{2}=100-64=36$ and $c=6$. Therefore the Foci lie at $(5,{-2}+6)$ and $(5,{-2}-6)$, or otherwise stated, at $(5,4)$ and $(5,{-8})$. Congratulations, you've completed all the elements.

### Part 2 Explanatory Charts, Diagrams, Photos

• (dependent upon the tutorial data above)
1. 1
Create the Excel Chart. Open a new workbook and start a new worksheet titled Ellipse.
• Input x into cell A1 and y into cell B1 and Format Cell font red, align center horizontal.
• Select between the A of Column A and the 1 of Row 1 and thus the entire worksheet at upper leftmost corner. Format Cells Number Number Decimal Places 4 and Font 9 or 10.
• Look over your figures to see the extent of the ellipse to guide your x values series. The Minor Axis runs from $({-3},2)$ to $(13,{-2})$ so you want to create a series of x values that range from -3 to 13. Select cell A2 and input -3 and select a cell about 40 cells down, say in A43, and input 13. Do Edit Fill Series Column Linear Trend OK.
• For the y values of the ellipse, you need to restate the ellipse's formula in terms of what y equals all by itself. Switch the y term with the 1 term and arrive at ${\frac {(x-5)^{2}}{64}}-1={\frac {-(y+2)^{2}}{100}}$. Multiply both sides by -100 and take the square root to arrive at ${\sqrt {100-{\frac {100}{64}}*(x-5)^{2}}}=y+2$. Bring the 2 to the left side to isolate y and arrive finally at ${\sqrt {100-{\frac {100}{64}}*(x-5)^{2}}}-2=y$. You can simplify ${\frac {100}{64}}$ to ${\frac {25}{16}}$ if you like -- Excel doesn't care much either way. Input that formula to B2 as =SQRT(100-(25/16)*(A2-5)^2)-2 where A2 is the X value, and select cell range B2:B43 and Edit Fill Down.
• Notice that almost all the y values are positive. Clearly, you've only taken half the square roots. Copy A2:A43 to A44 and paste and then sort the column from largest values to smallest. That sets up a continuum with the 13 ending at A43 and decreases around back to -3 for the x values. For the y values, copy the formula in cell B43 and insert a minus sign in front of it so that it reads now =-SQRT(100-(25/16)*(A44-5)^2)-2 when you copy and paste it into cell B44 to B85.
• Select A2:B85 and using the Chart Wizard or the Ribbon, select Charts All/Other, Scatter, Smooth Line Scatter and leave the chart on the Ellipse worksheet. Add vertical grid lines and stretch the chart until the grid squares are approximate squares.
• Go to cell B2 and select the formula in the formula bar and copy it and then select cell C2 and type y and paste the formula in after it. Make it bold.
• Go to cell D4 and type in the main formula and format font size 18 bold.
• Copy cells A1:B1 to cells D6:E6 for the x y headers in red.
• Input the Vertices $(x,y)$ pairs in cells D7:E8 and add the title Vertices (along Major Axis) in cell F7.
• Input the Co-Vertices $(x,y)$ pairs in cells D10:E11 and add the title Co-Vertices (along Minor Axis) in cell F10.
• Input the Center $(x,y)$ pair in cells D13:E13 and add the title Center in cell F13.
• Input the Foci $(x,y)$ pairs in cells D15:E16 and add the title Foci in cell F15.
• Select in the Plot Area of the Chart and do menu item Chart Add Data and in response to the data query/ies, respond with the $(x,y)$ pairs you just input. They may not "land correctly" and so require editing in the formula bar -- just edit them until they appear as given above, eg. Series 2 should be =SERIES("Vertices",Ellipse!$D$7:$D$8,Ellipse!$E$7:$E$8,2) and notice how "Vertices" was typed in the title at the beginning of the series description -- do that for each series so that they appear correctly in the Legend.
• Format the Vertices with a thin brown line and Size 8 red markers, or blue ones for the Co-Vertices. Format the Center as a large Light Blue Dot and the Foci as Large Black squares.
• Add data labels with both x and y if you prefer, as the chart above demonstrates, though it gets a bit crowded. Use font size 8 or 9.
• Move the Legend to the top of the Chart.
• Copy the Chart and Data with Grabber and save paste to a Paster worksheet to save your work and then save the workbook. 