Algebra.Com's Answer #827814 by ikleyn(52786)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A straight line L1 is reflected in the mirror line y=2x to give
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document.write( "the image L2 whose equation is y=1/2 x+2. Find the equation of L1.
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document.write( "Give your answer in the form ax+by=c where a, b and c are integers
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document.write( "First, let's find the intersection point of the mirror line y = 2x\r\n" );
document.write( "and line L2 whose equation is y = (1/2)x+2.\r\n" );
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document.write( "For it, we should solve the system of two equations\r\n" );
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document.write( " y = 2x,\r\n" );
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document.write( " y = 0.5x + 2.\r\n" );
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document.write( "It quickly reduces to \r\n" );
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document.write( " 2x = 0.5x + 2,\r\n" );
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document.write( "which gives the solution\r\n" );
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document.write( " 1.5x = 2, x =  =  =  .\r\n" );
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document.write( "Thus the mirror line and L2 intersect at the point with x-coordinate and y-coordinate  =  .\r\n" );
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document.write( "Again, the intersection point of the mirror line and L2 is the point (x,y) = (,).\r\n" );
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document.write( "It means that line L1 also passes through this point (it is the reason why we found this point).\r\n" );
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document.write( " +----------------------------------------------------------------+\r\n" );
document.write( " | At this point, half of the problem is just solved. |\r\n" );
document.write( " | From this point, the other half of the solution starts. |\r\n" );
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document.write( "The mirror line y = 2x has the slope 2; it means that its angle \"a\" with x-axis is tan(a) = 2.\r\n" );
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document.write( "Line L2 y = (1/2)x+2 has the slope 1/2 = 0.5; it means that its angle \"b\" with x-axis is tan(b) = 0.5.\r\n" );
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document.write( "Let's find the angle (a-b) between these lines. We have \r\n" );
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document.write( " tan(a-b) =  =  =  = 0.75 =  .\r\n" );
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document.write( "After mirroring about y = 2x, line L2 becomes L1 with the angle with x-axis a+(a-b) = 2a-b.\r\n" );
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document.write( "I want to calculate tan(2a-b), since it gives me the slope of line L1.\r\n" );
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document.write( "I calculate tan(2a) first: it is tan(2a) =  =  =  .\r\n" );
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document.write( "Next, I calculate tan(2a-b). It is\r\n" );
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document.write( " tan(2a-b) =  =  =  .\r\n" );
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document.write( "In the denominator, we have 1-1 = 0; it means that line L1 is vertical.\r\n" );
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document.write( "Since line L1 is vertical and passes through the point (,), its equation is\r\n" );
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document.write( " x = ,\r\n" );
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document.write( "or\r\n" );
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document.write( " 3x = 4.\r\n" );
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document.write( "ANSWER. An equation of line L1 in the requested form is 3x = 4, or (which is the same) 3x + 0*y = 4.\r\n" );
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document.write( "Solved.\r
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document.write( "This problem is of a Math Circle level.\r
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