document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039422>Question 1039422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> P (2p,p^2) is a variable point on the parabola x^2 = 4y. N is the foot of the perpendicular form P to the x axis NR is perpendicular to OP.\r<BR>\n" );
document.write( "\n" );
document.write( "a. find the equation of OP<BR>\n" );
document.write( "b. find the equation of NR<BR>\n" );
document.write( "c. show that the R has coordinates (8p/p^2 +4 , 4p^2/p^2+4)<BR>\n" );
document.write( "d. show that the locus R is a circle and state its centre and radius\r<BR>\n" );
document.write( "\n" );
document.write( "NEED HELP WITH PART C AND D!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654251 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=654251\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. {{OP}}} connect <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%282p%2Cp%5E2%29 ALIGN=MIDDLE  ALT=\"P%282p%2Cp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so its slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5E2%2F2p=p%2F2 ALIGN=MIDDLE  ALT=\"p%5E2%2F2p=p%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and its y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OP ALIGN=MIDDLE  ALT=\"OP\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28p%2F2%29x ALIGN=MIDDLE  ALT=\"y=%28p%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b. The foot of the perpendicular form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%282p%2Cp%5E2%29 ALIGN=MIDDLE  ALT=\"P%282p%2Cp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the x-axis is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%282p%2C0%29 ALIGN=MIDDLE  ALT=\"N%282p%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and the slope of a line perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OP ALIGN=MIDDLE  ALT=\"OP\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F%28p%2F2%29=-2%2Fp ALIGN=MIDDLE  ALT=\"-1%2F%28p%2F2%29=-2%2Fp\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=NR ALIGN=MIDDLE  ALT=\"NR\"   DISABLED_style= \"max-width:90%; height:auto;\" > in point-slope form, based on point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N%282p%2C0%29 ALIGN=MIDDLE  ALT=\"N%282p%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-2%2Fp%29%28x-2p%29 ALIGN=MIDDLE  ALT=\"y=%28-2%2Fp%29%28x-2p%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "c. you did not tell me where point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > is, but from its coordinates, I see that is a point on line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=OP ALIGN=MIDDLE  ALT=\"OP\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "so, R must be the intersecion of NR and OP.<BR>\n" );
document.write( "Hence, its coordinates are the solution to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=%28p%2F2%29x%2Cy=%28-2%2Fp%29%28x-2p%29%29 ALIGN=MIDDLE  ALT=\"system%28y=%28p%2F2%29x%2Cy=%28-2%2Fp%29%28x-2p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Using substitution, we start trying to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2%2Fp%29%28x-2p%29=%28p%2F2%29x ALIGN=MIDDLE  ALT=\"%28-2%2Fp%29%28x-2p%29=%28p%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Multiplying both sides times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2p ALIGN=MIDDLE  ALT=\"-2p\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-2p%29%28-2%2Fp%29%28x-2p%29=%28-2p%29%28p%2F2%29x ALIGN=MIDDLE  ALT=\"%28-2p%29%28-2%2Fp%29%28x-2p%29=%28-2p%29%28p%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28x-2p%29=-p%5E2x ALIGN=MIDDLE  ALT=\"4%28x-2p%29=-p%5E2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-8p=-p%5E2x ALIGN=MIDDLE  ALT=\"4x-8p=-p%5E2x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%2Bp%5E2x=8p ALIGN=MIDDLE  ALT=\"4x%2Bp%5E2x=8p\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Bp%5E2%29x=8p ALIGN=MIDDLE  ALT=\"%284%2Bp%5E2%29x=8p\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=8p%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28x=8p%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Then, substituting the value found for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28p%2F2%29x ALIGN=MIDDLE  ALT=\"y=%28p%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28p%2F2%29%288p%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"y=%28p%2F2%29%288p%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=4p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"highlight%28y=4p%5E2%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "d. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%288p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"R%288p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is symmetrical with respect to the y-axis,<BR>\n" );
document.write( "because substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-p ALIGN=MIDDLE  ALT=\"-p\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > you get point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%28-8p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"S%28-8p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the reflection of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%288p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"R%288p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > across the y-axis.<BR>\n" );
document.write( "Making <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=0 ALIGN=MIDDLE  ALT=\"p=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > gives us the point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which is part of the locus or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "All the points <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%288p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"R%288p%2F%284%2Bp%5E2%29%2C4p%5E2%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > have a non-negative y-coordinate,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4p%5E2%2F%284%2Bp%5E2%29%3E=0 ALIGN=MIDDLE  ALT=\"y=4p%5E2%2F%284%2Bp%5E2%29%3E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So the circle should be centered on the y-axis,<BR>\n" );
document.write( "pass though <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=O%280%2C0%29 ALIGN=MIDDLE  ALT=\"O%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and otherwise be above the y-axis.<BR>\n" );
document.write( "Such a circle has a radius r, and is centered at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%280%2Cr%29 ALIGN=MIDDLE  ALT=\"C%280%2Cr%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "All we need to do is find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation of such a circle would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-0%29%5E2%2B%28y-r%29%5E2=r%5E2 ALIGN=MIDDLE  ALT=\"%28x-0%29%5E2%2B%28y-r%29%5E2=r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2-2ry%2Br%5E2=r%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2-2ry%2Br%5E2=r%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2-2ry=0 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2-2ry=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=2ry ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=2ry\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=8p%2F%284%2Bp%5E2%29 ALIGN=MIDDLE  ALT=\"x=8p%2F%284%2Bp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=4p%5E2%2F%284%2Bp%5E2%29 ALIGN=MIDDLE  ALT=\"y=4p%5E2%2F%284%2Bp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we can find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%288p%2F%284%2Bp%5E2%29%29%5E2%2B%284p%5E2%2F%284%2Bp%5E2%29%29%5E2+=+2r%284p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64p%5E2%2F%284%2Bp%5E2%29%5E2%2B16p%5E4%2F%284%2Bp%5E2%29%5E2+=+8rp%5E2%2F%284%2Bp%5E2%29 ALIGN=MIDDLE  ALT=\"64p%5E2%2F%284%2Bp%5E2%29%5E2%2B16p%5E4%2F%284%2Bp%5E2%29%5E2+=+8rp%5E2%2F%284%2Bp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64p%5E2%2B16p%5E4+=+8rp%5E2%284%2Bp%5E2%29 ALIGN=MIDDLE  ALT=\"64p%5E2%2B16p%5E4+=+8rp%5E2%284%2Bp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16p%5E2%284%2Bp%5E2%29+=+8rp%5E2%284%2Bp%5E2%29 ALIGN=MIDDLE  ALT=\"16p%5E2%284%2Bp%5E2%29+=+8rp%5E2%284%2Bp%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16p%5E2=+8rp%5E2 ALIGN=MIDDLE  ALT=\"16p%5E2=+8rp%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16p%5E2%2F8p%5E2=r ALIGN=MIDDLE  ALT=\"16p%5E2%2F8p%5E2=r\"   DISABLED_style= \"max-width:90%; height:auto;\" > ---> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28r=2%29 ALIGN=MIDDLE  ALT=\"highlight%28r=2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The locus of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a circle of radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > centered at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%280%2C2%29 ALIGN=MIDDLE  ALT=\"C%280%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTES:<BR>\n" );
document.write( "The equations to graph that circle: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=8p%2F%284%2Bp%5E2%29%2Cy=4p%5E2%2F%284%2Bp%5E2%29%29 ALIGN=MIDDLE  ALT=\"system%28x=8p%2F%284%2Bp%5E2%29%2Cy=4p%5E2%2F%284%2Bp%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "can be called parametric equations.<BR>\n" );
document.write( "Parametric equations allow us to show a relation that is not a function,<BR>\n" );
document.write( "so we can express a relation that graphs as a curve we call a parametric curve.<BR>\n" );
document.write( "They are very useful if would be difficult to express with just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A third variable called parameter (in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) is uaed,<BR>\n" );
document.write( "and we define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > as functionc of that parameter.<BR>\n" );
document.write( "Many times, we try to \"eliminate the parameter\",<BR>\n" );
document.write( "to get to a relation between just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "be able to understand the relation between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > more easily.<BR>\n" );
document.write( "One way to do that is to solve one equation for the parameter,<BR>\n" );
document.write( "getting an expression for the parameter in terms of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > that can be substituted for the parameter in the other equation for the other variable. <BR>\n" );
document.write( "That would have been complicated in this case, but fortunately we were told it was a circle and all we had to do is prove it and find center and radius.<BR>\n" );
document.write( "That made it easier.<BR>\n" );
document.write( "Maybe the way I went about it is not the expected way, but it was the easiest way I could figure out.<BR>\n" );
document.write( "If you find out about some easier way, let me know.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );