document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=310821>Question 310821</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the equation of a line that is tangent to the circle (x-2)2 + y2= 25 at the point (6,3)?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #222329 by </B><A HREF=/tutors/aboutme.mpl?userid=Earlsdon><B>Earlsdon(6294)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Earlsdon><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=222329\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the tangent to the circle<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-2%29%5E2%2By%5E2+=+25 ALIGN=MIDDLE  ALT=\"%28x-2%29%5E2%2By%5E2+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the point (6,3)<BR>\n" );
document.write( "Using implicit differentiation, we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x-2%29%2A%281%29%2B2y%2A%28dy%2Fdx%29+=+0 ALIGN=MIDDLE  ALT=\"2%28x-2%29%2A%281%29%2B2y%2A%28dy%2Fdx%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Solve for the tangent (slope), <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx ALIGN=MIDDLE  ALT=\"dy%2Fdx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+-%28x-2%29%2Fy ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+-%28x-2%29%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" > and, at the point (6,3) the slope is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+-4%2F3 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-4%2F3 ALIGN=MIDDLE  ALT=\"m+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now find the equation of the line with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-4%2F3 ALIGN=MIDDLE  ALT=\"m+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and containing the point (6,3).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+mx%2Bb ALIGN=MIDDLE  ALT=\"y+=+mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-4%2F3 ALIGN=MIDDLE  ALT=\"m+=+-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3 ALIGN=MIDDLE  ALT=\"y+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6 ALIGN=MIDDLE  ALT=\"x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+-%284%2F3%29%2A6%2Bb ALIGN=MIDDLE  ALT=\"3+=+-%284%2F3%29%2A6%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify and solve for b.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+-8%2Bb ALIGN=MIDDLE  ALT=\"3+=+-8%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+11 ALIGN=MIDDLE  ALT=\"b+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The final equation, in slope-intercept form is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y+=+-%284%2F3%29x%2B11%29 ALIGN=MIDDLE  ALT=\"highlight%28y+=+-%284%2F3%29x%2B11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----------------------------------<BR>\n" );
document.write( "But there is a simpler way to do this!<BR>\n" );
document.write( "Find the equation of the line that is perpendicular to the radius of the circle and which contains the point (6,3).<BR>\n" );
document.write( "Recall that the tangent to a circle is perpendicular to the radius at the point of tangency.<BR>\n" );
document.write( "Find the slope of the radius whose end points are the circle center at (2,0) and the point of tangency at (6,3).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%283-0%29%2F%286-2%29 ALIGN=MIDDLE  ALT=\"m+=+%283-0%29%2F%286-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+3%2F4 ALIGN=MIDDLE  ALT=\"m+=+3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "The line that's perpendicular to this has a slope that's the negative reciprocal of<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F4 ALIGN=MIDDLE  ALT=\"3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F3 ALIGN=MIDDLE  ALT=\"-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so you can start with...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-%284%2F3%29x%2Bb ALIGN=MIDDLE  ALT=\"y+=+-%284%2F3%29x%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now substitute the x- and y-coordinates of the point of tangency (6,3).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+-%284%2F3%29%2A6%2Bb ALIGN=MIDDLE  ALT=\"3+=+-%284%2F3%29%2A6%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify and solve for b.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3+=+-8%2Bb ALIGN=MIDDLE  ALT=\"3+=+-8%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > so...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+11 ALIGN=MIDDLE  ALT=\"b+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "The equation is then...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight_green%28y+=+-%284%2F3%29x%2B11%29 ALIGN=MIDDLE  ALT=\"highlight_green%28y+=+-%284%2F3%29x%2B11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );