document.write( "Question 617296: Use the information provided to write the standard form of the ellipse:
\n" ); document.write( "Vertices: (-9,3),(-9,-23)
\n" ); document.write( "Foci: (-9,2), (-9,-22)
\n" ); document.write( "Please provide a step by step answer. \n" ); document.write( "
Algebra.Com's Answer #388296 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
  1. Since the vertices and foci all have the same x coordinates, the major axis must be a vertical line. The standard form for an ellipse with a vertical major axis is:
    \n" ); document.write( "\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2+=+1\"
  2. The center of the ellipse is halfway between the vertices (or foci). If you cannot \"see\" what halfway is, then just average the x coordinates and average the y coordinates. These averages will be the x and y coordinates of the center. You should find that the center is:
    \n" ); document.write( "(-9, -10)
    \n" ); document.write( "The coordinates in the center are represented by the \"h\" and \"k\" in the standard form.
  3. The \"a\" in the standard form is the distance from the center to either vertex. If you cannot \"see\" what this is, then use the distance formula to find the distance from the center to either vertex. You should get 13 for \"a\".
  4. The distance from the center to either focus is called \"c\". (\"c\" is not in the standard form but we need it to find \"b\".) Again, if you cannot see what this distance is, use the distance formula. You should find that \"c\" is 12.
  5. There is an equation that connects a, b and c in an ellipse:
    \n" ); document.write( "\"a%5E2+=+b%5E2+%2B+c%5E2\"
    \n" ); document.write( "Inserting our values for a and c we get:
    \n" ); document.write( "\"13%5E2+=+b%5E2+%2B+12%5E2\"
    \n" ); document.write( "which we can solve for b. Simplifying:
    \n" ); document.write( "\"169+=+b%5E2+%2B+144\"
    \n" ); document.write( "Subtracting 144:
    \n" ); document.write( "\"25+=+b%5E2\"
    \n" ); document.write( "Find the square root. (The a, b and c values are always considered positive numbers so we will ignore the negative square root of 25.)
    \n" ); document.write( "\"5+=+b\"
  6. We now have the h, k, a and b to fill into our standard form. (See #1):
    \n" ); document.write( "\"%28x-%28-9%29%29%5E2%2F5%5E2%2B%28y-%28-10%29%29%5E2%2F13%5E2+=+1\"
    \n" ); document.write( "which simplifies to:
    \n" ); document.write( "\"%28x%2B9%29%5E2%2F25%2B%28y%2B10%29%5E2%2F169+=+1\"
P.S. I notice that another tutor has solved this problem, too. But she thinks it is a horizontal ellipse. If you're not sure who is right, then plot the vertices and foci on a graph so see if they line up vertically as I say they do, or horizontally like she says. \n" ); document.write( "
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