document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1051725>Question 1051725</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the range of values of k for which kx+y=3 meets x²+y²=5 in two distinct points </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #667209 by </B><A HREF=/tutors/aboutme.mpl?userid=Fombitz><B>Fombitz(32388)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fombitz><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=Fombitz><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=667209\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> All the lines go through the point (0,3).<BR>\n" );
document.write( "The line tangent to the circle and going through the point (0,3) would make a right triangle together with the radius of the circle with the hypotenuse being from the origin to the point (0,3).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E2=5-D%5E2 ALIGN=MIDDLE  ALT=\"3%5E2=5-D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D=2 ALIGN=MIDDLE  ALT=\"D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So then build a circle centered at (0,3) with a radius of 2.<BR>\n" );
document.write( "Then look for the two intersection points of that circle with the original circle. <BR>\n" );
document.write( "Those two points are the two tangents points that the lines would make with the original circle and will provide the limits for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The equation of the new circle would be,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28y-3%29%5E2=2%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28y-3%29%5E2=2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%28y-3%29%5E2=4 ALIGN=MIDDLE  ALT=\"x%5E2%2B%28y-3%29%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So using the original equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=5 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=5-y%5E2 ALIGN=MIDDLE  ALT=\"x%5E2=5-y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substitute this into the new equation,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5-y%5E2%2B%28y-3%29%5E2=4 ALIGN=MIDDLE  ALT=\"5-y%5E2%2B%28y-3%29%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5-y%5E2%2By%5E2-6y%2B9=4 ALIGN=MIDDLE  ALT=\"5-y%5E2%2By%5E2-6y%2B9=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6y%2B14=4 ALIGN=MIDDLE  ALT=\"-6y%2B14=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6y=-10 ALIGN=MIDDLE  ALT=\"-6y=-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%2F3 ALIGN=MIDDLE  ALT=\"y=5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So them.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B%285%2F3%29%5E2=5 ALIGN=MIDDLE  ALT=\"x%5E2%2B%285%2F3%29%5E2=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2B25%2F9=5 ALIGN=MIDDLE  ALT=\"x%5E2%2B25%2F9=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=45%2F9-25%2F9 ALIGN=MIDDLE  ALT=\"x%5E2=45%2F9-25%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=20%2F9 ALIGN=MIDDLE  ALT=\"x%5E2=20%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0+%2B-+%282%2F3%29sqrt%285%29 ALIGN=MIDDLE  ALT=\"x=0+%2B-+%282%2F3%29sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So now you have two points on each line, you can solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kx%2By=3 ALIGN=MIDDLE  ALT=\"kx%2By=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%282%2F3%29sqrt%285%29%2B5%2F3=3 ALIGN=MIDDLE  ALT=\"k%282%2F3%29sqrt%285%29%2B5%2F3=3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%282%2F3%29sqrt%285%29=9%2F3-5%2F3 ALIGN=MIDDLE  ALT=\"k%282%2F3%29sqrt%285%29=9%2F3-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k%282%2F3%29sqrt%285%29=4%2F3 ALIGN=MIDDLE  ALT=\"k%282%2F3%29sqrt%285%29=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"k=2%2Fsqrt%285%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=%282%2F5%29sqrt%285%29 ALIGN=MIDDLE  ALT=\"k=%282%2F5%29sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with the other solution for k being the reciprocal,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=-%282%2F5%29sqrt%285%29 ALIGN=MIDDLE  ALT=\"k=-%282%2F5%29sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So the tangent points are the limit for one distinct point of contact so for two distinct points don't include these points so,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%282%2F5%29sqrt%285%29%3Ck%3C%282%2F5%29sqrt%285%29 ALIGN=MIDDLE  ALT=\"-%282%2F5%29sqrt%285%29%3Ck%3C%282%2F5%29sqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( ".<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=Fombitz&illustration_name=dfc3.JPG\">.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );