Algebra.Com's Answer #692320 by ikleyn(52803)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
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document.write( "there exist two circles that go through two points (1,3); (2,4) and are tangent to the y-axis.
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document.write( "Letting the radii of the circles be a, b implies that ab=?
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document.write( "*do not graph
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document.write( "The solution would be much easier to understand having a plot.
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document.write( "But, since you directly and explicitly ask do not graph, I will not use it.\r
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document.write( "1. Draw (mentally) the segment connecting the given points (1,3) and (2,4).\r\n" );
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document.write( " This segment has the slope 1 =  .\r\n" );
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document.write( " The centers of the two circles lie in the perpendicular bisector to this segment.\r\n" );
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document.write( " The perpendicular bisector goes through the middle point (1.5,3.5) and has the slope -1.\r\n" );
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document.write( "2. Let us find the radius of the \"upper\" circle.\r\n" );
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document.write( " Let \"p\" be the distance along the perpendicular bisector from the middle point (1.5,3.5) to the center of the \"upper\" circle.\r\n" );
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document.write( " Then the center of the \"upper\" circle is at the point ( , ), and the radius of the upper circle is  .\r\n" );
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document.write( " Since the upper circle touches y-axis, it gives the equation for \"p\"\r\n" );
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document.write( " = .\r\n" );
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document.write( " From this equation, p = (square both sides; simplify; then apply the quadratic formula) =  .\r\n" );
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document.write( " Then the radius of the upper circle is  =  = 1, which gives a = 1. (1)\r\n" );
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document.write( "3. Now, let us find the radius of the \"lower\" circle.\r\n" );
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document.write( " Let \"q\" be the distance along the perpendicular bisector from the middle point (1.5,3.5) to the center of the \"lower\" circle.\r\n" );
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document.write( " Then the center of the \"lower\" circle is at the point ( , ), and the radius of the lower circle is  .\r\n" );
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document.write( " Since the lower circle touches y-axis, it gives the equation for \"q\"\r\n" );
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document.write( " = .\r\n" );
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document.write( " From this equation, q = (square both sides; simplify; then apply the quadratic formula) =  .\r\n" );
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document.write( " Then the radius of the upper circle is  =  =  = 25, which gives b =  = 5. (2)\r\n" );
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document.write( "3. From (1) and (2), a*b = 1*5 = 5.\r\n" );
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document.write( "Answer. The product a*b under the question is 5.\r
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document.write( "Now, finally, I use all that I got and illustrate these results in the plot below.\r
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document.write( "It is my CHECK !\r
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document.write( "*** SOLVED ! ***\r
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