document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1158804>Question 1158804</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 9x^2-24xy+16y^2-20x-15y-50=0\r<BR>\n" );
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document.write( "Use axis rotation formulas for x and y to transform the quadratic equation to an equation in (u,v) coordinates with no cross product term. Identify the vertex or vertices in (x,y) coordinates\r<BR>\n" );
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document.write( "thank you so so so much </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854656 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5398)</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=854656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A quadratic equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0 ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-24xy%2B16y%5E2-20x-15y-50=0 ALIGN=MIDDLE  ALT=\"9x%5E2-24xy%2B16y%5E2-20x-15y-50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , could represent a circle, ellipse, hyperbola, parabola.<BR>\n" );
document.write( "In special cases it could represent a point, a line, a pair of lines, or no point that could exist, depending on the values of the coefficients.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ROTATING AXES:<BR>\n" );
document.write( "Axis rotation equations:<BR>\n" );
document.write( "From coordinates (x, y), to get the new coordinates (u, v) that would result for a point (x,y) after rotating the coordinate axes counterclockwise an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+alpha+ ALIGN=MIDDLE  ALT=\"+alpha+\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can apply<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=x%2Acos%28alpha%29%2By%2Asin%28alpha%29 ALIGN=MIDDLE  ALT=\"u=x%2Acos%28alpha%29%2By%2Asin%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=-x%2Asin%28alpha%29%2By%2Acos%28alpha%29 ALIGN=MIDDLE  ALT=\"v=-x%2Asin%28alpha%29%2By%2Acos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "From (u, v), rotating an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-alpha ALIGN=MIDDLE  ALT=\"-alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the reverse change) we get (x,y) using<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=u%2Acos%28-alpha%29-v%2Asin%28-alpha%29 ALIGN=MIDDLE  ALT=\"x=u%2Acos%28-alpha%29-v%2Asin%28-alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x=u%2Acos%28alpha%29-v%2Asin%28alpha%29%29 ALIGN=MIDDLE  ALT=\"highlight%28x=u%2Acos%28alpha%29-v%2Asin%28alpha%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=y=-u+%2Asin%28-alpha%29%2Bv%2Acos%28-alpha%29%29 ALIGN=MIDDLE  ALT=\"y=-u+%2Asin%28-alpha%29%2Bv%2Acos%28-alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29 ALIGN=MIDDLE  ALT=\"highlight%28y=u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%2Acos%28alpha%29-v%2Asin%28alpha%29 ALIGN=MIDDLE  ALT=\"u%2Acos%28alpha%29-v%2Asin%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > 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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29 ALIGN=MIDDLE  ALT=\"+u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   DISABLED_style= \"max-width:90%; height:auto;\" > in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-24xy%2B16y%5E2-20x-15y-50=0 ALIGN=MIDDLE  ALT=\"9x%5E2-24xy%2B16y%5E2-20x-15y-50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with a lot of busy algebra work, we could get a new equation in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > with new expressions for the new coefficients.<BR>\n" );
document.write( "That is grueling, mistake-inducing work (and torture to type and edit).<BR>\n" );
document.write( "We would have expressions involving <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all the new coefficients of terms with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and/or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > , including the coefficient of the term in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=uv ALIGN=MIDDLE  ALT=\"uv\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That coefficient must be made equal to zero.<BR>\n" );
document.write( "From the equation making the coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=uv ALIGN=MIDDLE  ALT=\"uv\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to zero, a value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > needs to be found to be used to calculate all other coefficients.<BR>\n" );
document.write( "Maybe that is the initial work that was expected for this problem.<BR>\n" );
document.write( " <BR>\n" );
document.write( "IDENTIFYING VERTEX (OR VERTICES):<BR>\n" );
document.write( "The new equation on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > should be identified as a parabola (one vertex), a hyperbola (two vertices), an ellipse (two vertices and two co-vertices), or something else.<BR>\n" );
document.write( "Then, the (u,v) coordinates for any vertices should be found, and converted into (x,y) coordinates.<BR>\n" );
document.write( " <BR>\n" );
document.write( "POSSIBLE SHORTCUTS:<BR>\n" );
document.write( "Rotating the coordinate axes counterclockwise an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > , from an equation <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0 ALIGN=MIDDLE  ALT=\"Ax%5E2%2BBxy%2BCy%5E2%2BDx%2BEy%2BF=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get a new equation in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > with new coefficients:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Jx%5E2%2BKxy%2BLy%5E2%2BMx%2BNy%2BF=0 ALIGN=MIDDLE  ALT=\"Jx%5E2%2BKxy%2BLy%5E2%2BMx%2BNy%2BF=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The new coefficients can be found to be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=J=A%2Acos%5E2%28alpha%29+%2BB%2Asin%28alpha%29cos%28alpha%29%2BC%2Asin%5E2%28alpha%29+ ALIGN=MIDDLE  ALT=\"J=A%2Acos%5E2%28alpha%29+%2BB%2Asin%28alpha%29cos%28alpha%29%2BC%2Asin%5E2%28alpha%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=K=+B%2A%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29%2B2%28C-A%29+sin%28alpha%29cos%28alpha%29%7D%29%7D%7D+%2C%0D%0A%7B%7B%7BL=+A%2Asin%5E2%28alpha%29+-B%2Asin%28alpha%29cos%28alpha%29%2BC%2Acos%5E2%28alpha%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=M=+D%2Acos%28alpha%29%2BE%2Asin%28alpha%29 ALIGN=MIDDLE  ALT=\"M=+D%2Acos%28alpha%29%2BE%2Asin%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=N=-D%2A+sin%28alpha%29%2BE%2Acos%28alpha%29 ALIGN=MIDDLE  ALT=\"N=-D%2A+sin%28alpha%29%2BE%2Acos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( "To eliminate the term in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=uv ALIGN=MIDDLE  ALT=\"uv\"   DISABLED_style= \"max-width:90%; height:auto;\" > we must find a value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > that makes<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2A%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29%2B2%28C-A%29+sin%28alpha%29cos%28alpha%29=0%7D%29%7D%7D%EF%83%A0+%7B%7B%7BB%2A%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29=2%28A-C%29sin%28alpha%29cos%28alpha%29%7D%29%7D%7D+.%0D%0AThat+equation+above+can+be+re-written+as+a+simpler+one+based+on+angle+%7B%7B%7B2alpha ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > using the trigonometric identities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282alpha%29=2+sin%28alpha%29cos%28alpha%29%7D%29%7D%7Dand+%7B%7B%7Bcos%282alpha%29=+cos%5E2%28alpha%29-sin%5E2%28alpha%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, it becomes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%2Acos%282alpha%29=%28A-C%29%2Asin%282alpha%29 ALIGN=MIDDLE  ALT=\"B%2Acos%282alpha%29=%28A-C%29%2Asin%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+B%2F%28A-C%29=sin%282alpha%29%2F+cos%282alpha%29 ALIGN=MIDDLE  ALT=\"+B%2F%28A-C%29=sin%282alpha%29%2F+cos%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28B%2F%28A-C%29=X%2C+sin%282alpha%29=cos%282alpha%29%28B%2F%28A-C%29%29%29 ALIGN=MIDDLE  ALT=\"system%28B%2F%28A-C%29=X%2C+sin%282alpha%29=cos%282alpha%29%28B%2F%28A-C%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . B/(A-C)=tan(2alpha), sin(2alpha)=cos(2alpha)(B/(A-C)<BR>\n" );
document.write( "Avoiding mistakes, with careful algebra work, I might have been able to find the expressions outlined above.<BR>\n" );
document.write( "I just copied them from an old Analytical Geometry textbook.<BR>\n" );
document.write( "The value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC ALIGN=MIDDLE  ALT=\"B%5E2-4AC\"   DISABLED_style= \"max-width:90%; height:auto;\" > , called the discriminant, suggests parabola, ellipse, or hyperbola if it is zero, negative, or positive respectively.<BR>\n" );
document.write( "I got  that from the book too.<BR>\n" );
document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2-24xy%2B16y%5E2-20x-15y-50=0 ALIGN=MIDDLE  ALT=\"9x%5E2-24xy%2B16y%5E2-20x-15y-50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC=24%5E2-4%2A9%2A16=3%5E2%2A8%5E2-9%2A8%5E2=576-576=0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC=24%5E2-4%2A9%2A16=3%5E2%2A8%5E2-9%2A8%5E2=576-576=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > says the equation represents a parabola.<BR>\n" );
document.write( " <BR>\n" );
document.write( "A SHORTCUT THAT SHOULD BE ALLOWED:<BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=u%2Acos%28alpha%29-v%2Asin%28alpha%29%2C+y=u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , we can translate into (u, v) coordinates the terms of degree 2 (the first 3 terms):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2=9%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2=9%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+red%289u%5E2%2Acos%5E2%28alpha%29%29-18uv%2Asin%28alpha%29cos%28alpha%29%2Bblue%289v%5E2%2Asin%5E2%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24xy=-24%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24%28u%5E2%2Acos%28alpha%29sin%28alpha%29+%2Buv%2Acos%5E2%28alpha%29+-uv%2Asin%5E2%28alpha%29+-v%5E2%2Asin%28alpha%29cos%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28-24u%5E2%2Acos%28alpha%29sin%28alpha%29%29-24uv%2Acos%5E2%28alpha%29%2B24uv%2Asin%5E2%28alpha%29%2Bblue%2824v%5E2%2Asin%28alpha%29cos%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28-24u%5E2%2Acos%28alpha%29sin%28alpha%29%29-24uv%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29%2Bblue%2824v%5E2%2Asin%28alpha%29cos%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16y%5E2=16%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29%5E2 ALIGN=MIDDLE  ALT=\"16y%5E2=16%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%2816u%5E2%2A+sin%5E2%28alpha%29%29%2B32uv%2Asin%28alpha%29cos%28alpha%29%2Bblue%2816v%5E2%2Acos%5E2%28alpha%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " <BR>\n" );
document.write( "When “collecting like terms”, the term in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=uv ALIGN=MIDDLE  ALT=\"uv\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18uv%2Asin%28alpha%29cos%28alpha%29+%2B32uv%2Asin%28alpha%29cos%28alpha%29 ALIGN=MIDDLE  ALT=\"-18uv%2Asin%28alpha%29cos%28alpha%29+%2B32uv%2Asin%28alpha%29cos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . <BR>\n" );
document.write( "Making the coefficient of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=uv ALIGN=MIDDLE  ALT=\"uv\"   DISABLED_style= \"max-width:90%; height:auto;\" > equal to zero, we get the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-18sin%28alpha%29cos%28alpha%29+%2B32sin%28alpha%29cos%28alpha%29=0 ALIGN=MIDDLE  ALT=\"-18sin%28alpha%29cos%28alpha%29+%2B32sin%28alpha%29cos%28alpha%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "We can simplify that equation using the trigonometric identities<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282alpha%29=cos%5E2%28alpha%29-sin%5E2%28alpha%29 ALIGN=MIDDLE  ALT=\"cos%282alpha%29=cos%5E2%28alpha%29-sin%5E2%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282alpha%29=2sin%28alpha%29cos%28alpha%29 ALIGN=MIDDLE  ALT=\"sin%282alpha%29=2sin%28alpha%29cos%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7sin%282alpha%29=24cos%282alpha%29 ALIGN=MIDDLE  ALT=\"7sin%282alpha%29=24cos%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28sin%282alpha%29=%2824%2F7%29cos%282alpha%29%2C+and%2C+tan%282alpha%29=24%2F7%29 ALIGN=MIDDLE  ALT=\"system%28sin%282alpha%29=%2824%2F7%29cos%282alpha%29%2C+and%2C+tan%282alpha%29=24%2F7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > From that we would not get an exact value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2alpha ALIGN=MIDDLE  ALT=\"2alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > , but we know that alpha is in.<BR>\n" );
document.write( "A calculator would give you an approximation with enough digits to figure out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=0.8 ALIGN=MIDDLE  ALT=\"cos%28alpha%29=0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%29=0.6 ALIGN=MIDDLE  ALT=\"sin%28alpha%29=0.6\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We can get exact values of the trigonometric functions and their squares, from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282alpha%29=%2824%2F7%29cos%282alpha%29 ALIGN=MIDDLE  ALT=\"sin%282alpha%29=%2824%2F7%29cos%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%5E2%282alpha%29%2Bcos%5E2%282alpha%29=1 ALIGN=MIDDLE  ALT=\"sin%5E2%282alpha%29%2Bcos%5E2%282alpha%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2824%2F7%29cos%282alpha%29%29%5E2%2Bcos%5E2%282alpha%29=1 ALIGN=MIDDLE  ALT=\"%28%2824%2F7%29cos%282alpha%29%29%5E2%2Bcos%5E2%282alpha%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28576%2F49%29cos%5E2%282alpha%29%2Bcos%5E2%282alpha%29=1 ALIGN=MIDDLE  ALT=\"%28576%2F49%29cos%5E2%282alpha%29%2Bcos%5E2%282alpha%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28576%2F49%2B1%29cos%5E2%282alpha%29=1 ALIGN=MIDDLE  ALT=\"%28576%2F49%2B1%29cos%5E2%282alpha%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28625%2F49%29cos%5E2%282alpha%29=1 ALIGN=MIDDLE  ALT=\"%28625%2F49%29cos%5E2%282alpha%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%5E2%282alpha%29=49%2F625 ALIGN=MIDDLE  ALT=\"cos%5E2%282alpha%29=49%2F625\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cos%282alpha%29=7%2F25%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%282alpha%29=7%2F25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cos%282alpha%29=7%2F25=cos%5E2%28alpha%29-sin%5E2%28alpha%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%282alpha%29=7%2F25=cos%5E2%28alpha%29-sin%5E2%28alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282alpha%29=%2824%2F7%29cos%282alpha%29 ALIGN=MIDDLE  ALT=\"sin%282alpha%29=%2824%2F7%29cos%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%282alpha%29=%2824%2F7%29%287%2F25%29=24%2F25 ALIGN=MIDDLE  ALT=\"sin%282alpha%29=%2824%2F7%29%287%2F25%29=24%2F25\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sin%282alpha%29=24%2F25=2sin%28alpha%29cos%28alpha%29%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%282alpha%29=24%2F25=2sin%28alpha%29cos%28alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The sine and cosine of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > can also be calculated from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%282alpha%29 ALIGN=MIDDLE  ALT=\"cos%282alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > using the trigonometric identities for half angles as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%29=sqrt%28%281-cos%282alpha%29%29%2F2%29=sqrt%28%281-7%2F25%29%2F2%29=sqrt%289%2F25%29=3%2F5=0.6 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sin%28alpha%29=3%2F5=0.6%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%28alpha%29=3%2F5=0.6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=sqrt%28%281%2Bcos%282alpha%29%29%2F2%29=sqrt%28%281%2B7%2F25%29%2F2%29=sqrt%2816%2F25%29=4%2F5=0.8 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cos%28alpha%29=4%2F5=0.8%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%28alpha%29=4%2F5=0.8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Now we can go back to expressions for the terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2 ALIGN=MIDDLE  ALT=\"u%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E2 ALIGN=MIDDLE  ALT=\"v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and substitute for the trigonometric functions involved the values we found and highlighted before.<BR>\n" );
document.write( "As found before they come from the original terms <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-24xy ALIGN=MIDDLE  ALT=\"-24xy\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16y%5E2 ALIGN=MIDDLE  ALT=\"16y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > expressed as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha ALIGN=MIDDLE  ALT=\"alpha\"   DISABLED_style= \"max-width:90%; height:auto;\" > : <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28sin%282alpha%29=24%2F25-sin%28alpha%29cos%28alpha%29%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%282alpha%29=24%2F25-sin%28alpha%29cos%28alpha%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%28cos%282alpha%29=7%2F25=cos%5E2%28alpha%29-sin%5E2%28alpha%29%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%282alpha%29=7%2F25=cos%5E2%28alpha%29-sin%5E2%28alpha%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%28sin%28alpha%29=3%2F5=0.6%29 ALIGN=MIDDLE  ALT=\"highlight%28sin%28alpha%29=3%2F5=0.6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28cos%28alpha%29=4%2F5=0.8%29 ALIGN=MIDDLE  ALT=\"highlight%28cos%28alpha%29=4%2F5=0.8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9x%5E2=9%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29%5E2 ALIGN=MIDDLE  ALT=\"9x%5E2=9%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+red%289u%5E2%2Acos%5E2%28alpha%29%29-18uv%2Asin%28alpha%29cos%28alpha%29%2Bblue%289v%5E2%2Asin%5E2%28alpha%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=-24xy ALIGN=MIDDLE  ALT=\"-24xy\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28-24u%5E2%2Acos%28alpha%29sin%28alpha%29%29-24uv%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29%2Bblue%2824v%5E2%2Asin%28alpha%29cos%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16y%5E2=16%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29%5E2 ALIGN=MIDDLE  ALT=\"16y%5E2=16%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%2816u%5E2%2A+sin%5E2%28alpha%29%29%2B32uv%2Asin%28alpha%29cos%28alpha%29%2Bblue%2816v%5E2%2Acos%5E2%28alpha%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Collecting like terms, we find the term in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u%5E2 ALIGN=MIDDLE  ALT=\"u%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%289u%5E2%2Acos%5E2%28alpha%29%29-red%2824u%5E2%2Acos%28alpha%29sin%28alpha%29%29%2Bred%2816u%5E2%2A+sin%5E2%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%289%284%2F5%29%5E2%28alpha%29-24%284%2F5%29%283%2F5%29%2B16%2A%283%2F5%29%5E2%29u%5E2 ALIGN=MIDDLE  ALT=\"red%289%284%2F5%29%5E2%28alpha%29-24%284%2F5%29%283%2F5%29%2B16%2A%283%2F5%29%5E2%29u%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%289%2A16%2F25-12%2812%2F25%29%2B16%2A%289%2F25%29%29u%5E2 ALIGN=MIDDLE  ALT=\"red%289%2A16%2F25-12%2812%2F25%29%2B16%2A%289%2F25%29%29u%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28%28144-288-1440%2F25%29%29u%5E2 ALIGN=MIDDLE  ALT=\"red%28%28144-288-1440%2F25%29%29u%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28red%280%29%2Au%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28red%280%29%2Au%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the tern in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E2 ALIGN=MIDDLE  ALT=\"v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%289v%5E2%2Asin%5E2%28alpha%29%29%2Bblue%2824v%5E2%2Asin%28alpha%29cos%28alpha%29%29%2Bblue%2816v%5E2%2Acos%5E2%28alpha%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%289%283%2F5%29%5E2%2B24%283%2F5%29%284%2F5%29%2B16%284%2F5%29%5E2%29v%5E2 ALIGN=MIDDLE  ALT=\"blue%289%283%2F5%29%5E2%2B24%283%2F5%29%284%2F5%29%2B16%284%2F5%29%5E2%29v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%289%2A9%2F25%2B24%2A12%2F25%2B16%2816%2F25%29%29v%5E2 ALIGN=MIDDLE  ALT=\"blue%289%2A9%2F25%2B24%2A12%2F25%2B16%2816%2F25%29%29v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%2881%2B288%2B256%2F25%29%29v%5E2 ALIGN=MIDDLE  ALT=\"blue%2881%2B288%2B256%2F25%29%29v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28625%2F25%29%29v%5E2 ALIGN=MIDDLE  ALT=\"blue%28625%2F25%29%29v%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28blue%2825%29v%5E2%29 ALIGN=MIDDLE  ALT=\"highlight%28blue%2825%29v%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "The linear (degree 1) terms in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > come from<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20x=-20%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29 ALIGN=MIDDLE  ALT=\"-20x=-20%28u%2Acos%28alpha%29-v%2Asin%28alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20%28%284%2F5%29u-%283%2F5%29v%29=-16u%2B12v ALIGN=MIDDLE  ALT=\"-20%28%284%2F5%29u-%283%2F5%29v%29=-16u%2B12v\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15y=-15%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29 ALIGN=MIDDLE  ALT=\"-15y=-15%28u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-15%28%283%2F5%29u%2B%284%2F5v%29%29=-9u-12v ALIGN=MIDDLE  ALT=\"-15%28%283%2F5%29u%2B%284%2F5v%29%29=-9u-12v\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-20x-15v=-16u%2B12v-9u-12v=-25u%2B0v ALIGN=MIDDLE  ALT=\"-20x-15v=-16u%2B12v-9u-12v=-25u%2B0v\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The equation on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v ALIGN=MIDDLE  ALT=\"v\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%280%29%2Au%5E2%2B0%2Auv%2Bblue%2825%29v%5E2-25u%2B0v-50=0 ALIGN=MIDDLE  ALT=\"red%280%29%2Au%5E2%2B0%2Auv%2Bblue%2825%29v%5E2-25u%2B0v-50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25v%5E2-25u-50=0 ALIGN=MIDDLE  ALT=\"25v%5E2-25u-50=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which simplifies to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E2-u-2=0 ALIGN=MIDDLE  ALT=\"v%5E2-u-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=v%5E2-2 ALIGN=MIDDLE  ALT=\"u=v%5E2-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , representing a parabola with axis <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v=0 ALIGN=MIDDLE  ALT=\"v=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and vertex <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%28-2%2C0%29 ALIGN=MIDDLE  ALT=\"V%28-2%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > all in u,v coordinates.<BR>\n" );
document.write( "The x,y coordinates of the vertex are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=u%2Acos%28alpha%29-v%2Asin%28alpha%29=-2%2A%284%2F5%29-0%2A%283%2F5%29=-8%2F5=-1.6 ALIGN=MIDDLE  ALT=\"x=u%2Acos%28alpha%29-v%2Asin%28alpha%29=-2%2A%284%2F5%29-0%2A%283%2F5%29=-8%2F5=-1.6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29=-2%2A%283%2F5%29%2B0%2A%284%2F5%29=6%2F5=-1.2 ALIGN=MIDDLE  ALT=\"y=u%2Asin%28alpha%29%2Bv%2Acos%28alpha%29=-2%2A%283%2F5%29%2B0%2A%284%2F5%29=6%2F5=-1.2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );