document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=750389>Question 750389</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please Explain as much as you can I really need help:<BR>\n" );
document.write( "Find an equation of the ellipse having the given points as foci and the given sum of the focal radii<BR>\n" );
document.write( "(-9, 0); (9, 0); 30 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #456558 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=456558\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28ellipse%29 ALIGN=MIDDLE  ALT=\"red%28ellipse%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is sort of a stretched <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28circumference%29 ALIGN=MIDDLE  ALT=\"blue%28circumference%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-5.5%2C5.5%2C-5.5%2C5.5%2C%0D%0Ablue%28circle%280%2C2%2C3%29%29%2Cblue%28circle%280%2C2%2C0.1%29%29%2Clocate%28-0.7%2C2%2Ccenter%29%2C%0D%0Ared%28circle%28-sqrt%287%29%2C-3.5%2C0.1%29%29%2Cred%28circle%28sqrt%287%29%2C-3.5%2C0.1%29%29%2C%0D%0Alocate%28-3.2%2C-3.5%2Cfocus%29%2Clocate%282%2C-3.5%2Cfocus%29%2C%0D%0Ared%28ellipse%280%2C-3.5%2C8%2C3%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "A GARDENER STORY:<BR>\n" );
document.write( "If a gardener wanted to mark/draw a circular flower bed, he/she would stick a stake in the ground, and holding that loop tight around another stake, would draw a the circle in the ground with the free stake. The fixed stake, stuck in the ground, is the center of the circle. The distance from the free stake to the fixed one is constant and is the radius of the circle.<BR>\n" );
document.write( " <BR>\n" );
document.write( "For an ellipse, the gardener sticks two stakes in the ground and draws the ellipse the same way using a third stake. The sum of the distances between all 3 stakes is constant. The fixed stakes are the foci of the ellipse. The distance between them is constant. The distances from the fixed stakes (foci) to the other stake (a point on the ellipse) is what your teacher calls the focal radii. Those distances change as the ellipse is being drawn, but their sum is constant. To draw the ellipse in yout problem you set your stakes (foci) on the x-axis), at a distance of 9 units to either side of the origin. The length of your loop of string should be the 18 unit distance between the foci, plus the 30 units sum of the focal radii. <BR>\n" );
document.write( " <BR>\n" );
document.write( "FACTS YOU NEED TO KNOW:<BR>\n" );
document.write( "The midpoint of the segment connecting the foci is the center of the ellipse, and in this case its is (0,0).<BR>\n" );
document.write( "The line containing the foci contains the major axis (the longest dimension) of the ellipse, including the vertices (the two ends of the major axis).<BR>\n" );
document.write( "In this case, the line containing the foci, major axis and vertices is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the x-axis. <BR>\n" );
document.write( "An ellipse centered at the origin with the major axis along the x-axis has an equation of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= semi-major axis and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >= semi-minor axis<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the horizontal distance from the center to each end (vertex) to the left and right. Those left and right ends are called the vertices.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the vertical distance from the center up and down to the top and bottom of the ellipse. (those two points are called the co-vertices. There is also a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >= focal distance (distance from each focus to the center).<BR>\n" );
document.write( "In your problem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=9 ALIGN=MIDDLE  ALT=\"c=9\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The relanships below let you calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >= sum of the focal radii (in the problem <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=30 ALIGN=MIDDLE  ALT=\"2a=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=15 ALIGN=MIDDLE  ALT=\"a=15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%5E2%2F15%5E2%2By%5E2%2F12%5E2=1%29 ALIGN=MIDDLE  ALT=\"highlight%28x%5E2%2F15%5E2%2By%5E2%2F12%5E2=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%5E2%2F225%2By%5E2%2F144=1%29 ALIGN=MIDDLE  ALT=\"highlight%28x%5E2%2F225%2By%5E2%2F144=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "SOLUTION:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=9 ALIGN=MIDDLE  ALT=\"c=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a=30 ALIGN=MIDDLE  ALT=\"2a=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=15 ALIGN=MIDDLE  ALT=\"a=15\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%5E2=b%5E2%2B9%5E2 ALIGN=MIDDLE  ALT=\"15%5E2=b%5E2%2B9%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=225=b%5E2%2B81 ALIGN=MIDDLE  ALT=\"225=b%5E2%2B81\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2=225-81=144 ALIGN=MIDDLE  ALT=\"b%5E2=225-81=144\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=12 ALIGN=MIDDLE  ALT=\"b=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE:<BR>\n" );
document.write( "Since it is customary to make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E=b ALIGN=MIDDLE  ALT=\"a%3E=b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, If the major axis was along the y-, and the left and right ends would be co-vertices. Also, since it is customary to makw <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3Eb ALIGN=MIDDLE  ALT=\"a%3Eb\"   DISABLED_style= \"max-width:90%; height:auto;\" >, they would switch the {{a}}} and {{b}}}, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2 ALIGN=MIDDLE  ALT=\"a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > under the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2 ALIGN=MIDDLE  ALT=\"y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but otherwise the formula would be the same).<BR>\n" );
document.write( " <BR>\n" );
document.write( "REASONS BEHIND FACTS:<BR>\n" );
document.write( "Memorizing formulas may help you pass a course. Knowing the reasoning behind the formula means you do not have to memorize it, and you will remember it when faced with the same concept in a course next year (or years from now).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C240%2C-16.5%2C16.5%2C-13.2%2C13.2%2C%0D%0Aline%28-17%2C0%2C17%2C0%29%2Cline%280%2C-15%2C0%2C15%29%2C%0D%0Ared%28circle%28-9%2C0%2C0.5%29%29%2Cred%28circle%289%2C0%2C0.5%29%29%2C%0D%0Alocate%28-9.5%2C-0.5%2C%22F%27%22%29%2Clocate%288.5%2C-0.5%2CF%29%2C%0D%0Ablue%28circle%2815%2C0%2C0.5%29%29%2Cblue%28circle%280%2C12%2C0.5%29%29%2C%0D%0Alocate%2815.2%2C-0.5%2CP%29%2Clocate%2801%2C12%2CQ%29%2C%0D%0Alocate%280.3%2C-0.2%2CO%29%2Cred%28ellipse%280%2C0%2C30%2C24%29%29%2C%0D%0Ablue%28line%28-9%2C0%2C0%2C12%29%29%2Cblue%28line%289%2C0%2C0%2C12%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > OP =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > OQ =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, OF = F'O =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "F'P = F'O + OP =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Ba ALIGN=MIDDLE  ALT=\"c%2Ba\"   DISABLED_style= \"max-width:90%; height:auto;\" > and FP = OP - FP =<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a-c ALIGN=MIDDLE  ALT=\"a-c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Sum of focal radii for point P = F'P + FP = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28c%2Ba%29%2B%28a-c%29=2a ALIGN=MIDDLE  ALT=\"%28c%2Ba%29%2B%28a-c%29=2a\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so<BR>\n" );
document.write( "Sum of focal radii for all points on the ellipse = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a ALIGN=MIDDLE  ALT=\"2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Sum of focal radii for point Q = F'Q + FQ = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2FQ+=+2a ALIGN=MIDDLE  ALT=\"2FQ+=+2a\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=FQ=a ALIGN=MIDDLE  ALT=\"FQ=a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the right triangle OPQ, the Pythagorean theorem says:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=FQ%5E2=OQ%5E2%2BOF%5E2 ALIGN=MIDDLE  ALT=\"FQ%5E2=OQ%5E2%2BOF%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2=b%5E2%2Bc%5E2 ALIGN=MIDDLE  ALT=\"a%5E2=b%5E2%2Bc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );