document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1001766>Question 1001766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the intersection of a cone and a plane parallel to a line along the side of the cone? \r<BR>\n" );
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document.write( "Provide mathematical examples to support your opinions. You may use equations, diagrams, or graphs to organize and present your thoughts. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619065 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=619065\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The person asking the question has a meaning in mind for the expressions \"a cone\" and \"a plane parallel to a line along the side of the cone\".<BR>\n" );
document.write( "Not knowing him/her, I cannot read that mind, so I must make some assumptions.\r<BR>\n" );
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document.write( "A straight line along the side of the cone must go through the vertex of the cone.<BR>\n" );
document.write( "Let's place that cone on a 3D x-y-z coordinate space,<BR>\n" );
document.write( "or rather, I will place x-, y-, and z-axes around that cone.<BR>\n" );
document.write( "I will place the vertex at the origin,<BR>\n" );
document.write( "the axis of the cone along the positive y-axis, and<BR>\n" );
document.write( "the \"line along the side of the cone\" on the x-y plane.<BR>\n" );
document.write( "The cross section of the cone with the line looks kind of like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-0.5%2C0.5%2C-0.1%2C0.9%2Cabs%282x%29%2B0.01%2C2x-0.01%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-0.5%2C0.5%2C-0.1%2C0.9%2Cabs%282x%29%2B0.01%2C2x-0.01%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I assume that by cone we mean an infinite lateral surface,<BR>\n" );
document.write( "with no base, or a base represented by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=infinity ALIGN=MIDDLE  ALT=\"y=infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" > , if you wish.<BR>\n" );
document.write( "I am not assuming that the infinite cone extends to any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3C0 ALIGN=MIDDLE  ALT=\"y%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation of the line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx ALIGN=MIDDLE  ALT=\"y=mx\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The equation for the cone would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Asqrt%28x%5E2%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"y=m%2Asqrt%28x%5E2%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the z-axis is the axis that we do not see,<BR>\n" );
document.write( "because it comes perpendicularly out of the screen towards us,<BR>\n" );
document.write( "but at every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=k ALIGN=MIDDLE  ALT=\"y=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > level, the \"horizontal\" section of the cone is a circle,<BR>\n" );
document.write( "centered at the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=0%2Cy=k%2Cz=0%29 ALIGN=MIDDLE  ALT=\"system%28x=0%2Cy=k%2Cz=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <BR>\n" );
document.write( "with radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" >, such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%5E2=x%5E2%2Bz%5E2 ALIGN=MIDDLE  ALT=\"R%5E2=x%5E2%2Bz%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=sqrt%28x%5E2%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"R=sqrt%28x%5E2%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "Now, how could we place a plane parallel to that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx ALIGN=MIDDLE  ALT=\"y=mx\"   DISABLED_style= \"max-width:90%; height:auto;\" > line?\r<BR>\n" );
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document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx ALIGN=MIDDLE  ALT=\"y=mx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is part of the plane <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=mx%2Cz=anything%29 ALIGN=MIDDLE  ALT=\"system%28y=mx%2Cz=anything%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > that is tangent to the cone's surface.<BR>\n" );
document.write( "A plane parallel to that plane, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=mx%2Bb%2Cz=anything%29 ALIGN=MIDDLE  ALT=\"system%28y=mx%2Bb%2Cz=anything%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "is a plane that I would call parallel to the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx ALIGN=MIDDLE  ALT=\"y=mx\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is probably what was envisioned in the question. I assume so.<BR>\n" );
document.write( "The intersection of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=mx%2Bb%2Cz=anything%29 ALIGN=MIDDLE  ALT=\"system%28y=mx%2Bb%2Cz=anything%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Asqrt%28x%5E2%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"y=m%2Asqrt%28x%5E2%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is obviously<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=mx%2Bb=m%2Asqrt%28x%5E2%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"mx%2Bb=m%2Asqrt%28x%5E2%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28mx%2Bb%29%5E2=m%5E2%28x%5E2%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"%28mx%2Bb%29%5E2=m%5E2%28x%5E2%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5E2x%5E2%2Bb%5E2%2B2mbx=m%5E2x%5E2%2Bm%5E2z2%29 ALIGN=MIDDLE  ALT=\"m%5E2x%5E2%2Bb%5E2%2B2mbx=m%5E2x%5E2%2Bm%5E2z2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2B2mbx=m%5E2z2%29 ALIGN=MIDDLE  ALT=\"b%5E2%2B2mbx=m%5E2z2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2mbx=m%5E2z2-b%5E2%29 ALIGN=MIDDLE  ALT=\"2mbx=m%5E2z2-b%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=m%5E2z2%2F2mb-b%5E2%2F2mb%29 ALIGN=MIDDLE  ALT=\"x=m%5E2z2%2F2mb-b%5E2%2F2mb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28m%2F2b%29%2Az2-b%2F2m%29 ALIGN=MIDDLE  ALT=\"x=%28m%2F2b%29%2Az2-b%2F2m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and that equation does look like a parabola.<BR>\n" );
document.write( "Of course, the intersection of the cone and plane is not on the x-z plane.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28m%2F2b%29%2Az2-b%2F2m%29 ALIGN=MIDDLE  ALT=\"x=%28m%2F2b%29%2Az2-b%2F2m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is just the projection of that parabolic intersection on the x-z plane.<BR>\n" );
document.write( "The actual intersection is slightly stretched version of that projection. <BR>\n" );
document.write( "We can keep the x-axis, but we need a new <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X ALIGN=MIDDLE  ALT=\"X\"   DISABLED_style= \"max-width:90%; height:auto;\" > axis.<BR>\n" );
document.write( "That is the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y=mx%2Bb%2Cz=0%29 ALIGN=MIDDLE  ALT=\"system%28y=mx%2Bb%2Cz=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and on that axis a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=1 ALIGN=MIDDLE  ALT=\"x=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > becomes a distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=DELTA ALIGN=MIDDLE  ALT=\"DELTA\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=X=m%5E2%2B1 ALIGN=MIDDLE  ALT=\"X=m%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the parabola is just a little stretched.\r<BR>\n" );
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document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx ALIGN=MIDDLE  ALT=\"y=mx\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also part of the plane <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=anything%2Cy=anything%2Cz=0%29 ALIGN=MIDDLE  ALT=\"system%28x=anything%2Cy=anything%2Cz=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > that contains the axis of the cone.<BR>\n" );
document.write( "A plane parallel to that plane, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=anything%2Cy=anything%2Cz=k%29 ALIGN=MIDDLE  ALT=\"system%28x=anything%2Cy=anything%2Cz=k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "is also a plane that I would call parallel to the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx ALIGN=MIDDLE  ALT=\"y=mx\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The intersection of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=anything%2Cy=anything%2Cz=k%29 ALIGN=MIDDLE  ALT=\"system%28x=anything%2Cy=anything%2Cz=k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Asqrt%28x%5E2%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"y=m%2Asqrt%28x%5E2%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is obviously <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=m%2Asqrt%28x%5E2%2Bk%5E2%29 ALIGN=MIDDLE  ALT=\"y=m%2Asqrt%28x%5E2%2Bk%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3E=0%2Cy%5E2=m%5E2%28x%5E2%2Bk%5E2%29%29 ALIGN=MIDDLE  ALT=\"system%28y%3E=0%2Cy%5E2=m%5E2%28x%5E2%2Bk%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3E=0%2Cy%5E2=m%5E2x%5E2%2Bm%5E2k%5E2%29 ALIGN=MIDDLE  ALT=\"system%28y%3E=0%2Cy%5E2=m%5E2x%5E2%2Bm%5E2k%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3E=0%2Cy%5E2-m%5E2x%5E2=m%5E2k%5E2%29 ALIGN=MIDDLE  ALT=\"system%28y%3E=0%2Cy%5E2-m%5E2x%5E2=m%5E2k%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28y%3E=0%2Cy%5E2%2Fm%5E2k%5E2-x%5E2%2Fk%5E2=1%29 ALIGN=MIDDLE  ALT=\"system%28y%3E=0%2Cy%5E2%2Fm%5E2k%5E2-x%5E2%2Fk%5E2=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That is half of a hyperbola on the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28x=anything%2Cy=anything%2Cz=k%29 ALIGN=MIDDLE  ALT=\"system%28x=anything%2Cy=anything%2Cz=k%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > plane.\r<BR>\n" );
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document.write( "There is an infinite number of other planes that I could consider,<BR>\n" );
document.write( "but I think of them as linear combinations of the planes already described. </SPAN>\n" );
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