document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1146382>Question 1146382</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equations of the lines passing through the origin that are tangent to a circle with radius 2 and center at point (2, 1). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #767656 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=767656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Clearly with a center (2,1) and radius 2, one of the two lines tangent to the circle passing through the origin is x=0.<br><BR>\n" );
document.write( "Some work is needed to find the other line....<br><BR>\n" );
document.write( "Let (a,b) be the other point of tangency to the given circle of a line that passes through the origin.  Then<br><BR>\n" );
document.write( "(1) The distance from (2,1) to (a,b) is 2:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b-1%29%5E2%2B%28a-2%29%5E2+=+2%5E2 ALIGN=MIDDLE  ALT=\"%28b-1%29%5E2%2B%28a-2%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=b%5E2-2b%2B1%2Ba%5E2-4a%2B4+=+4 ALIGN=MIDDLE  ALT=\"b%5E2-2b%2B1%2Ba%5E2-4a%2B4+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2-4a-2b+=+-1 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2-4a-2b+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) The slope of the radius to the point of tangency is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28b-1%29%2F%28a-2%29 ALIGN=MIDDLE  ALT=\"%28b-1%29%2F%28a-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slope of the tangent line is the negative reciprocal, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-a%29%2F%28b-1%29 ALIGN=MIDDLE  ALT=\"%282-a%29%2F%28b-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The tangent passes through the points (0,0) and (a,b); so an equation of the tangent is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%28%282-a%29%2F%28b-1%29%29a ALIGN=MIDDLE  ALT=\"b+=+%28%282-a%29%2F%28b-1%29%29a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28b-1%29+=+a%282-a%29 ALIGN=MIDDLE  ALT=\"b%28b-1%29+=+a%282-a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2-b+=+2a-a%5E2 ALIGN=MIDDLE  ALT=\"b%5E2-b+=+2a-a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2Bb%5E2-2a-b+=+0 ALIGN=MIDDLE  ALT=\"a%5E2%2Bb%5E2-2a-b+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Subtracting the equation in (1) from the equation in (2),<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2Bb+=+1 ALIGN=MIDDLE  ALT=\"2a%2Bb+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1-2a ALIGN=MIDDLE  ALT=\"b+=+1-2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Substituting b=1-2a in (2)...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281-2a%29%28-2a%29+=+2a-a%5E2 ALIGN=MIDDLE  ALT=\"%281-2a%29%28-2a%29+=+2a-a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2a%2B4a%5E2+=+2a-a%5E2 ALIGN=MIDDLE  ALT=\"-2a%2B4a%5E2+=+2a-a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5a%5E2-4a+=+0 ALIGN=MIDDLE  ALT=\"5a%5E2-4a+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%285a-4%29+=+0 ALIGN=MIDDLE  ALT=\"a%285a-4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+4%2F5 ALIGN=MIDDLE  ALT=\"a+=+4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And then, to find b<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+1-2a+=+-3%2F5 ALIGN=MIDDLE  ALT=\"b+=+1-2a+=+-3%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The point (a,b) is (4/5,-3/5).<br><BR>\n" );
document.write( "Since the tangent line passes through the origin, the equation of the line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%2F4%29x ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F4%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph showing part of the lower half of the given circle and the second tangent line; the first tangent line is of course x=0, the y-axis:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-1%2C3%2C-2%2C2%2C-sqrt%284-%28x-2%29%5E2%29%2B1%2C%28-3%2F4%29x%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-1%2C3%2C-2%2C2%2C-sqrt%284-%28x-2%29%5E2%29%2B1%2C%28-3%2F4%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );