document.write( "Question 721364: Write an equation for each ellipse.\r
\n" ); document.write( "\n" ); document.write( "Center at (-3,6); major axis vertical, with length 10; C=2 \n" ); document.write( "
Algebra.Com's Answer #442444 by jsmallt9(3759)\"\" \"About 
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The standard forms for the equation of a ellipse are:
\n" ); document.write( "\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\" for horizontal ellipses
\n" ); document.write( "and
\n" ); document.write( "\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\" for vertical ellipses
\n" ); document.write( "In these standard forms...
  • The \"h\" and \"k\" are the x and y coordinates of the center of the hyperbola.
  • The \"a\" is the distance from the center to a vertex on the major axis.
  • The \"b\" is the distance from the center to a vertex on the minor axis.

\n" ); document.write( "Since we've been told that the major axis is vertical, our ellipse is vertical. So from this point on, we will be using:
\n" ); document.write( "\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"
\n" ); document.write( "
  • We know that the center is (-3, 6). So the h and k of the equation will be -3 and 6.
  • Since the length of the major axis is 10, the distance from the center to each vertex will be half of that, or 5. So a = 5.
  • To find \"b\" we will use the equation that connects the values of a, b and c in an ellipse:
    \n" ); document.write( "\"a%5E2=b%5E2%2Bc%5E2\"
    \n" ); document.write( "\"%285%29%5E2=b%5E2%2B%282%29%5E2\"
    \n" ); document.write( "\"25=b%5E2%2B4\"
    \n" ); document.write( "\"21=b%5E2\"
    \n" ); document.write( "\"sqrt%2821%29=b\" (We ignore the negative square root since b is a distance.)
  • Now that we have the h, k, a and b values we can write the equation:
    \n" ); document.write( "\"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\"

    \n" ); document.write( "\"%28x-%28-3%29%29%5E2%2F%28sqrt%2821%29%29%5E2%2B%28y-%286%29%29%5E2%2F%285%29%5E2=1\"
    \n" ); document.write( "which simplifies to:
    \n" ); document.write( "\"%28x%2B3%29%5E2%2F21%2B%28y-6%29%5E2%2F25=1\" \n" ); document.write( "
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