document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164537>Question 1164537</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)Find a new representation of the given equation after rotating through the given angle.\r<BR>\n" );
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document.write( "11x^2-50√3xy-39y^2+576=0          60degrees </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854198 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5345)</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=854198\">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;\" > could represent a circle, ellipse, hyperbola, parabola. There are special cases where I 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" );
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document.write( "In the case of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11x%5E2-50sqrt%283%29xy-39y%5E2%2B576=0 ALIGN=MIDDLE  ALT=\"11x%5E2-50sqrt%283%29xy-39y%5E2%2B576=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we have<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S=11 ALIGN=MIDDLE  ALT=\"S=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=-50sqrt%283%29 ALIGN=MIDDLE  ALT=\"B=-50sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-39 ALIGN=MIDDLE  ALT=\"C=-39\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=574 ALIGN=MIDDLE  ALT=\"F=574\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=E=0 ALIGN=MIDDLE  ALT=\"D=E=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC=%28-50sqrt%283%29%29%5E2-4%2A11%2A%28-39%29=7500%2B1716=9216 ALIGN=MIDDLE  ALT=\"B%5E2-4AC=%28-50sqrt%283%29%29%5E2-4%2A11%2A%28-39%29=7500%2B1716=9216\"   DISABLED_style= \"max-width:90%; height:auto;\" > suggest it's a hyperbola.<BR>\n" );
document.write( "Symmetry and other considerations help visualize the curve represented too.<BR>\n" );
document.write( "We can see that if a point (x,y) satisfies that equation, the point (-x, -y) will also satisfy that equation. That tells us that the set of points satisfying that equation is symmetrical with respect to the origin.<BR>\n" );
document.write( "When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the equation turns into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-39y%5E2%2B576=0 ALIGN=MIDDLE  ALT=\"-39y%5E2%2B576=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and there is here is no solution 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;\" > , so we know that the curve does not touch or cross the y-axis. That, for an equation with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2 ALIGN=MIDDLE  ALT=\"x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <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;\" > makes me suspect an hyperbola.<BR>\n" );
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document.write( "Rotating such a curve (counterclockwise) by 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 would get another equation with the coefficients A', B', C', and F' replacing A, B, C, and F.<BR>\n" );
document.write( "The formulas to find the new coefficients are:<BR>\n" );
document.write( "A'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2Acos%5E2%28alpha%29%2BB%2Acos%28alpha%29%2Asin%28alpha%29%2BC%2Asin%5E2%28alpha%29 ALIGN=MIDDLE  ALT=\"A%2Acos%5E2%28alpha%29%2BB%2Acos%28alpha%29%2Asin%28alpha%29%2BC%2Asin%5E2%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "B'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29%2B2%28C-A%29cos%28alpha%29%2Asin%28alpha%29 ALIGN=MIDDLE  ALT=\"B%28cos%5E2%28alpha%29-sin%5E2%28alpha%29%29%2B2%28C-A%29cos%28alpha%29%2Asin%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "C'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2Asin%5E2%28alpha%29-B%2Acos%28alpha%29%2Asin%28alpha%29%2BC%2Acos%5E2%28alpha%29 ALIGN=MIDDLE  ALT=\"A%2Asin%5E2%28alpha%29-B%2Acos%28alpha%29%2Asin%28alpha%29%2BC%2Acos%5E2%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "F'=F<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=alpha=60%5Eo ALIGN=MIDDLE  ALT=\"alpha=60%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28alpha%29=1%2F2 ALIGN=MIDDLE  ALT=\"cos%28alpha%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28alpha%29=sqrt%283%29%2F2 ALIGN=MIDDLE  ALT=\"sin%28alpha%29=sqrt%283%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2A%281%2F4%29%2B%28-50sqrt%283%29%29%2A%281%2F2%29%2A%28sqrt%283%29%2F2%29%2B%28-39-11%29%2A%283%2F4%29%0D%0A=11%2F4-50%2A3%2F4-48%2A%283%2F4%29=%2811-150-117%29%2F4=-256%2F4=highlight%28-64%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "B'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-50sqrt%283%29%29%281%2F4-3%2F4%29%2B2%28-39-11%29%281%2F2%29%2A%28sqrt%283%29%2F2%29=%28-50sqrt%283%29%29%28-1%2F2%29%2B2%28-50%29%281%2F2%29%2A%28sqrt%283%29%2F2%29=50sqrt%283%29%2F2-50sqrt%283%29%2F2=highlight%280%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "C'=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2A%283%2F4%29-%28-50sqrt%283%29%29%2A%28sqrt%283%29%2F2%29%2A%281%2F2%29%2B%28-39%29%2A%281%2F4%29=%2833%2B150-39%29%2F4=144%2F4=highlight%2836%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "F'=576<BR>\n" );
document.write( "The rotated curve equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-64x%5E2%2B36y%5E2=576 ALIGN=MIDDLE  ALT=\"-64x%5E2%2B36y%5E2=576\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which can be rewritten as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F16-x%5E2%2F9=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F16-x%5E2%2F9=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2F4%5E2-x%5E2%2F3%5E2=1 ALIGN=MIDDLE  ALT=\"y%5E2%2F4%5E2-x%5E2%2F3%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That represents a hyperbola centered at the origin, with vertices at (0,-4) and (0,4) and asymptotes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%22+%22+%2B-+%284%2F3%29x ALIGN=MIDDLE  ALT=\"y=%22+%22+%2B-+%284%2F3%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Her is what the two branches of the hyperbola and its asymptotes look like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C320%2C-5%2C5%2C-8%2C8%2C4sqrt%281%2Bx%5E2%2F9%29%2C-4sqrt%281%2Bx%5E2%2F9%29%2C4x%2F3%2C-4x%2F3%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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