document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1143991>Question 1143991</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a set of parametric equations for the conic section or line below;<BR>\n" );
document.write( "Ellipse: Center: (5,-3); Vertices: 6 units above and below the center; Endpoint of minor Axis: 2 units left and right of center. \r<BR>\n" );
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document.write( "A. x=5+6cos t, y=-3+2 sin t<BR>\n" );
document.write( "B. x=5-2cos t, y=-3-6 sin t<BR>\n" );
document.write( "C. x=5+2cos t, y=-3+6 sin t<BR>\n" );
document.write( "D. x=2+5cos t, y=6-3 sin t </SPAN>\n" );
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document.write( "Let's start with a generic set of parametric equations and see how it relates to the equation of an ellipse in rectangular coordinates.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+a%2Bb%2Acos%28t%29 ALIGN=MIDDLE  ALT=\"x+=+a%2Bb%2Acos%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+c%2Bd%2Asin%28t%29 ALIGN=MIDDLE  ALT=\"y+=+c%2Bd%2Asin%28t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The parameter is t; eliminate it using sin^2+cos^2=1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28t%29+=+%28x-a%29%2Fb ALIGN=MIDDLE  ALT=\"cos%28t%29+=+%28x-a%29%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sint%28t%29+=+%28y-c%29%2Fd ALIGN=MIDDLE  ALT=\"sint%28t%29+=+%28y-c%29%2Fd\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28t%29%5E2%2Bcos%28t%29%5E2+=+%28x-a%29%5E2%2Fb%5E2%2B%28y-c%29%5E2%2Fd%5E2+=+1 ALIGN=MIDDLE  ALT=\"sin%28t%29%5E2%2Bcos%28t%29%5E2+=+%28x-a%29%5E2%2Fb%5E2%2B%28y-c%29%5E2%2Fd%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The center is (a,c); the semi-major and semi-minor axes are (in some order) b and d.<br><BR>\n" );
document.write( "Now use that analysis for the given ellipse.<br><BR>\n" );
document.write( "The center is (a,c) = (5,-3), so the constants in the parametric equations for x and y must be 5 and -3, respectively.  That eliminates answer choice D.<br><BR>\n" );
document.write( "The semi-major axis is 6, in the y direction, so d is 6.  That makes answer choice C the only possible correct answer.<br><BR>\n" );
document.write( "It can be seen that answer choice C is in fact correct, because it shows the semi-minor axis having length b=2.<br><BR>\n" );
document.write( "ANSWER: C. x=5+2cos t, y=-3+6 sin t </SPAN>\n" );
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