document.write( "Question 1043332: Determine the center,foci,vertices,and covertices of the ellipse with the given equation:\r
\n" ); document.write( "\n" ); document.write( "9x^2+16y^2-126x+64y=71 \n" ); document.write( "
Algebra.Com's Answer #658488 by KMST(5328)\"\" \"About 
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\"9x%5E2%2B16y%5E2-126x%2B64y=71\"
\n" ); document.write( "\"9x%5E2-126x%2B16y%5E2%2B64y=71\"
\n" ); document.write( "\"9%28x%5E2-14x%29%2B16%28y%5E2%2B4y%29=71\"
\n" ); document.write( "Thge expressions in brackets remindme of some squares:
\n" ); document.write( "\"x%5E2-14x%2B49=%28x-7%29%5E2\" , so \"9%28x-7%29%5E2%29=9%28x%5E2-14x%2B49%29=9x%5E2-126x%2B441\" , and
\n" ); document.write( "\"y%5E2%2B4y%2B4=%28y%2B2%29%5E2\" , so \"16%28y%2B2%29%5E2%29=16%28y%5E2%2B4y%2B4%29=16y%5E2%2B64y%2B64\" .
\n" ); document.write( "Adding \"441%2B64\" to both sides of the equal sign in the original equation, we have
\n" ); document.write( "\"9x%5E2%2B16y%5E2-126x%2B64y%2B441%2B64=71%2B441%2B64\"
\n" ); document.write( "\"9x%5E2-126x%2B441%2B16y%5E2%2B64y%2B441%2B64=576\"
\n" ); document.write( "\"9%28x%5E2-14x%2B49%29%2B16%28y%5E2%2B4y%2B4%29=576\"
\n" ); document.write( "\"9%28x-7%29%5E2%2B16%28y%2B2%29%5E2=576\"
\n" ); document.write( "Dividing both sides of the equal sign in the equation above by \"9%2A16\" , we have
\n" ); document.write( "\"%28x-7%29%5E2%2F16%2B%28y%2B2%29%5E2%2F9=576%2F%289%2A16%29\"
\n" ); document.write( "\"%28x-7%29%5E2%2F16%2B%28y%2B2%29%5E2%2F9=4\"
\n" ); document.write( "Dividing both sides of the equal sign in the equation above by \"4\" , we have
\n" ); document.write( "\"%28x-7%29%5E2%2F64%2B%28y%2B2%29%5E2%2F36=1\" , or \"%28x-7%29%5E2%2F8%5E2%2B%28y%2B2%29%5E2%2F6%5E1=1\" .
\n" ); document.write( "That is the equation of an ellipse centered at \"%22%28+7+%2C+-2+%29%22\" ,
\n" ); document.write( "the point with \"highlight%28system%28x=7%2Cy=-2%29%29\" .
\n" ); document.write( "The semi-major axis length is \"a=8\" , and
\n" ); document.write( "semi-minor axis length is \"b=6\" .
\n" ); document.write( "The \"a%5E2=8%5E2=64\" is dividing the term with \"x\",
\n" ); document.write( "so the major axis is parallel to the x-axis,
\n" ); document.write( "and since the center has \"y=-2\",
\n" ); document.write( "the major axis is the line \"y=-2\" .
\n" ); document.write( "On that major axis, are the vertices and foci.
\n" ); document.write( "The minor axis is parallel to the y-axis,
\n" ); document.write( "and since the center has \"x=7\",
\n" ); document.write( "the major axis is the line \"x=7\" .
\n" ); document.write( "We know that the focal distance of an ellipse, \"c\" , can be found as
\n" ); document.write( "\"c=sqrt%28a%5E2-b%5E2%29\" , so in this case
\n" ); document.write( "\"c=sqrt%2864-36%29=sqrt%2828%29=4sqrt%287%29\" . An approximate value is \"c=5.29\" .
\n" ); document.write( "The locations of the foci are on the major axis,
\n" ); document.write( "at a distance \"c=4sqrt%287%29=about5.29\" to either side of the center,
\n" ); document.write( "so one the coordinates of the foci are
\n" ); document.write( "\"highlight%28system%28x=7-4sqrt%287%29%2Cy=-2%29%29\" , or about \"%22%28+1.71+%2C+-2+%29%22\" for one focus,
\n" ); document.write( "and \"highlight%28system%28x=7%2B4sqrt%287%29%2Cy=-2%29%29\" , or about \"%22%28+12.29+%2C+-2+%29%22\" for the other focus.
\n" ); document.write( "Similarly, the vertices are on the major axis,
\n" ); document.write( "at a distance \"a=8\" to either side of the center,
\n" ); document.write( "so one the coordinates of the vertices are
\n" ); document.write( "\"system%28x=7-8%2Cy=-2%29=highlight%28system%28x=-1%2Cy=-2%29%29\" ,
\n" ); document.write( "and \"system%28x=7%2B8%2Cy=-2%29=highlight%28system%28x=15%2Cy=-2%29%29\" .
\n" ); document.write( "The covertices are on the minor axis,
\n" ); document.write( "at a distance \"b=6\" to either side of the center,
\n" ); document.write( "so one the coordinates of the vertices are
\n" ); document.write( "\"system%28x=7%2Cy=-2-6%29=highlight%28system%28x=7%2Cy=-8%29%29\" ,
\n" ); document.write( "and \"system%28x=7%2Cy=-2%2B6%29=highlight%28system%28x=7%2Cy=4%29%29\" .
\n" ); document.write( "The ellipse with axes and foci looks like this:
\n" ); document.write( " \n" ); document.write( "
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