document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1016364>Question 1016364</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The ellipse <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2%2F64+%2B+y%5E2%2F25+=+1 ALIGN=MIDDLE  ALT=\"+x%5E2%2F64+%2B+y%5E2%2F25+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is given. Find the equations of the lines tangent to this ellipse which make an angle of 60 degrees with the x-axis.\r<BR>\n" );
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document.write( "can someone show me how to do and answer it. thank you very much for help </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #632754 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=632754\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If the line makes an angle of 60 degrees with the x-axis, then one pair of lines would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+tan%2860%5Eo%29%2Ax+%2B+k ALIGN=MIDDLE  ALT=\"y+=+tan%2860%5Eo%29%2Ax+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%29x%2Bk ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29x%2Bk\"   DISABLED_style= \"max-width:90%; height:auto;\" > where k would have two values each of which is the negative of the other.<BR>\n" );
document.write( "Substitute this into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E2%2F64+%2B+y%5E2%2F25+=+1 ALIGN=MIDDLE  ALT=\"+x%5E2%2F64+%2B+y%5E2%2F25+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, from which we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F64+%2B+%28sqrt%283%29%2Ax%2Bk%29%5E2%2F25+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2F64+%2B+%28sqrt%283%29%2Ax%2Bk%29%5E2%2F25+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F64+%2B+%283x%5E2%2B2sqrt%283%29%2Akx+%2Bk%5E2%29%2F25+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2F64+%2B+%283x%5E2%2B2sqrt%283%29%2Akx+%2Bk%5E2%29%2F25+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "After simplifying, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28217%2F1600%29%2Ax%5E2+%2B+%28%282sqrt%283%29%29%2F25%29%2Akx+%2B+k%5E2%2F25-1+=+0 ALIGN=MIDDLE  ALT=\"%28217%2F1600%29%2Ax%5E2+%2B+%28%282sqrt%283%29%29%2F25%29%2Akx+%2B+k%5E2%2F25-1+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Now since the line is supposed to be tangent to the ellipse, the discriminant of the quadratic equation above must be 0.\r<BR>\n" );
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document.write( "Hence\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2F625%29%2Ak%5E2+-+4%2A%28217%2F1600%29%2A%28k%5E2%2F25+-1%29+=+0 ALIGN=MIDDLE  ALT=\"%2812%2F625%29%2Ak%5E2+-+4%2A%28217%2F1600%29%2A%28k%5E2%2F25+-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F400%29%2Ak%5E2+%2B+217%2F400+=+0 ALIGN=MIDDLE  ALT=\"%28-1%2F400%29%2Ak%5E2+%2B+217%2F400+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%5E2+=+217 ALIGN=MIDDLE  ALT=\"k%5E2+=+217\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+sqrt%28217%29 ALIGN=MIDDLE  ALT=\"k+=+sqrt%28217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k+=+-sqrt%28217%29 ALIGN=MIDDLE  ALT=\"k+=+-sqrt%28217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Thus the two lines are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%29x+%2B+sqrt%28217%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29x+%2B+sqrt%28217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+sqrt%283%29x+-+sqrt%28217%29 ALIGN=MIDDLE  ALT=\"y+=+sqrt%283%29x+-+sqrt%28217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Note that we cannot accept the other pair of lines <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%29x+%2B+sqrt%28217%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%29x+%2B+sqrt%28217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-sqrt%283%29x+-+sqrt%28217%29 ALIGN=MIDDLE  ALT=\"y+=+-sqrt%283%29x+-+sqrt%28217%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > because these two lines make an angle of 120 degrees with the x-axis.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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