document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191486>Question 1191486</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. What is the ratio of the area of the first circle to the sum of the areas of the other circles in the sequence? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823287 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</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=823287\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Here is a diagram of the largest circle with center P and the next circle with center O.<br><BR>\n" );
document.write( "The circles in the problem will have radii that form a decreasing geometric sequence; we need to find the ratio of that sequence.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-1%2C11%2C-1%2C11%0D%0A%2Cline%280%2C0%2C10%2C0%29%2Cline%280%2C0%2C0%2C10%29%0D%0A%2Ccircle%281%2C1%2C1%29%2Ccircle%285.828%2C5.828%2C5.828%29%0D%0A%2Cline%280%2C0%2C5.828%2C5.828%29%2Cline%285.828%2C0%2C5.828%2C5.828%29%2Cline%281%2C0%2C1%2C1%29%0D%0A%2Clocate%28.7%2C1.3%2CO%29%2Clocate%285.9%2C6.2%2CP%29%2Clocate%280%2C0%2CA%29%2Clocate%281%2C0%2CB%29%2Clocate%285.8%2C0%2CC%29%2Clocate%281.6%2C2.4%2CD%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Let the smaller circle have radius 1 and the larger circle have radius x.  We need to find the ratio 1/x.<br><BR>\n" );
document.write( "Then we have<BR>\n" );
document.write( "AB = OB = OD = 1<BR>\n" );
document.write( "AO = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%282%29 ALIGN=MIDDLE  ALT=\"sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "AC = PC = PD = x<br><BR>\n" );
document.write( "Triangle ACP has legs of length x and a hypotenuse of length <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "Therefore,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1%2Bsqrt%282%29=x%2Asqrt%282%29 ALIGN=MIDDLE  ALT=\"x%2B1%2Bsqrt%282%29=x%2Asqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Bsqrt%282%29=x%2Asqrt%282%29-x+=+x%28sqrt%282%29-1%29 ALIGN=MIDDLE  ALT=\"1%2Bsqrt%282%29=x%2Asqrt%282%29-x+=+x%28sqrt%282%29-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28sqrt%282%29%2B1%29%2F%28sqrt%282%29-1%29+=+%28%28sqrt%282%29%2B1%29%28sqrt%282%29%2B1%29%29%2F%28%28sqrt%282%29-1%29%28sqrt%282%29%2B1%29%29+=+%283%2B2sqrt%282%29%29%2F%282-1%29+=+3%2B2sqrt%282%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the radius of circle O is smaller than the radius of circle P by a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%283%2B2sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"1%2F%283%2B2sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That means the area of circle O is smaller than the area of circle P by a factor of<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F%283%2B2sqrt%282%29%29%29%5E2+=+1%2F%2817%2B12sqrt%282%29%29 ALIGN=MIDDLE  ALT=\"%281%2F%283%2B2sqrt%282%29%29%29%5E2+=+1%2F%2817%2B12sqrt%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That is the ratio of the geometric sequence of the areas of the ever smaller circles.  Use the formula for the sum of an infinite geometric series, using \"1\" as the area of the first circle.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+1%2F%281-r%29+=+1%2F%281-%281%2F%2817%2B12sqrt%282%29%29%29%29+=+1%2F%28%2817%2B12sqrt%282%29-1%29%2F%2817%2B12sqrt%282%29%29%29+=+%2817%2B12sqrt%282%29%29%2F%2816%2B12sqrt%282%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That number is the ratio of the sum of the areas of the infinite sequence of circles to the area of the first circle.<br><BR>\n" );
document.write( "The question asks for the ratio of the area of the first circle to the sum the areas of all the OTHER circles.  That ratio is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2816%2B12sqrt%282%29%29%2F%28%2817%2B12sqrt%282%29%29-%2816%2B12sqrt%282%29%29%29=16%2B12sqrt%282%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );