document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182794>Question 1182794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two circles, one of radius 1 and the other of radius 2, touch externally at P.<BR>\n" );
document.write( "A straight line through P cuts the area formed by these two circles in the ratio 1:2.<BR>\n" );
document.write( "In what ratio does this line cut the area of the smaller circle? </SPAN>\n" );
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document.write( "Here is a sketch of the two circles and the line through their point of tangency:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-3%2C5%2C-3%2C5%2Ccircle%28-1%2C0%2C1%29%2Ccircle%282%2C0%2C2%29%0D%0A%2Cline%28-1.8%2C-.5%2C3.6%2C1%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The figures of the small circle with its chord and of the large circle and its chord are similar figures.<br><BR>\n" );
document.write( "The radius of the small circle is 1; the radius of the large circle is 2.  So the areas of the two circles are pi and 4pi.<br><BR>\n" );
document.write( "Let x be the area of the smaller portion of the small circle; then the area of the larger portion is (pi-x).<br><BR>\n" );
document.write( "Since the scale factor between the two circles is 1:2, the ratio of corresponding area measurements between the two circles is 1:4.  So the smaller portion of the large circle is 4x, and the larger portion of the large circle is 4(pi-x) = (4pi-4x).<br><BR>\n" );
document.write( "We are told that the line divides the total area of the two circles in the ratio 1:2.  The total area is 5pi.  That means the area of the smaller portion of the small circle, plus the larger portion of the area of the large circle, is 2/3 of the total 5pi:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%2B%284pi-4x%29=%2810%2F3%29pi ALIGN=MIDDLE  ALT=\"%28x%29%2B%284pi-4x%29=%2810%2F3%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4pi-3x+=+%2810%2F3%29pi ALIGN=MIDDLE  ALT=\"4pi-3x+=+%2810%2F3%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+=+%282%2F3%29pi ALIGN=MIDDLE  ALT=\"3x+=+%282%2F3%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%282%2F9%29pi ALIGN=MIDDLE  ALT=\"x+=+%282%2F9%29pi\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The area of the smaller portion of the small circle is x=(2/9)pi; that means the area of the larger portion of the small circle is (pi-x) = (7/9)pi.<br><BR>\n" );
document.write( "And that means the smaller circle is divided in the ratio (2/9):(7/9) = 2:7<br><BR>\n" );
document.write( "ANSWER: 2:7<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );