document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=308992>Question 308992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4x^2 + ky^2 - 8x + 17y = 3\r<BR>\n" );
document.write( "\n" );
document.write( "Find the vaule of k to make this equation a circle, ellipse, hyperbola, and parabola; so different vaules for k that will make those different kinds of equations. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Any single one of them would be helpful if you can't necessairly figure them all out, so if you know anything at all pleaseee help. Thank you so much!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #220922 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=220922\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is very helpful to remember:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the general conic <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;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3C0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the given conic above is an ellipse<BR>\n" );
document.write( "Furthermore, if <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;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the conic is also a circle\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC=0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the given conic above is a parabola\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3E0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then the given conic above is a hyperbola\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First, let's subtract 3 from both sides to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+ky%5E2+-+8x+%2B+17y+-3=0 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+ky%5E2+-+8x+%2B+17y+-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So if we wanted to force <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+ky%5E2+-+8x+%2B+17y+-3=0 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+ky%5E2+-+8x+%2B+17y+-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be a circle, then we must make sure that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3C0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <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;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. In this case, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=4 ALIGN=MIDDLE  ALT=\"A=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=k ALIGN=MIDDLE  ALT=\"C=k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=-8 ALIGN=MIDDLE  ALT=\"D=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E=17 ALIGN=MIDDLE  ALT=\"E=17\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F=-3 ALIGN=MIDDLE  ALT=\"F=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Plug these values in to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%5E2-4%284%29%28k%29%3C0 ALIGN=MIDDLE  ALT=\"0%5E2-4%284%29%28k%29%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-16k%3C0 ALIGN=MIDDLE  ALT=\"-16k%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Solve for 'k' to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%3E0 ALIGN=MIDDLE  ALT=\"k%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So 'k' must be positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Also, because we want <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=0 ALIGN=MIDDLE  ALT=\"B=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=k ALIGN=MIDDLE  ALT=\"C=k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=k ALIGN=MIDDLE  ALT=\"A=k\"   DISABLED_style= \"max-width:90%; height:auto;\" > as well. But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=4 ALIGN=MIDDLE  ALT=\"A=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=4 ALIGN=MIDDLE  ALT=\"k=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This means that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=4 ALIGN=MIDDLE  ALT=\"k=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we get the circle  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2+%2B+4y%5E2+-+8x+%2B+17y+-3=0 ALIGN=MIDDLE  ALT=\"4x%5E2+%2B+4y%5E2+-+8x+%2B+17y+-3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For any ellipse, just pick a positive 'k' value that is NOT equal to 4. This 'k' value will make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3C0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > true.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the parabola, just make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=0 ALIGN=MIDDLE  ALT=\"k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > since this satisfies the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC=0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (basically, everything goes to zero since B and C are zero)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And finally, for any hyperbola, reverse the idea of the ellipse and pick any negative 'k' value. This works because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3E0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is essentially the opposite of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%5E2-4AC%3C0 ALIGN=MIDDLE  ALT=\"B%5E2-4AC%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );