document.write( "Question 1200960: The line (x/a) + (y/b) = 1, where a and b are positive constant, intersects the x- and y-axes at
\n" ); document.write( "the points A and B respectively. The mid-point of AB lies on the line 2x + y = 10 and
\n" ); document.write( "the distance AB = 10. Find the values of a and b. \n" ); document.write( "
Algebra.Com's Answer #835216 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "The line \"x%2Fa+%2B+y%2Fb+=+1\" is in intercept form with x-intercept (a,0) and \r\n" );
document.write( "y-intercept (0,b). So the midpoint of AB is \"%28matrix%281%2C3%2C%28a%2B0%29%2F2%2C%22%2C%22%2C%280%2Bb%29%2F2%29%29\" = \"%28matrix%281%2C3%2Ca%2F2%2C%22%2C%22%2Cb%2F2%29%29\".\r\n" );
document.write( "\r\n" );
document.write( "Since it lies on the line \"2x+%2B+y+=+10\", \r\n" );
document.write( "\r\n" );
document.write( "\"2%28a%2F2%29+%2B+%28b%2F2%29+=+10\",\r\n" );
document.write( "\r\n" );
document.write( "\"a%2Bb%2F2=10\"\r\n" );
document.write( "\"2a%2Bb=20\"\r\n" );
document.write( "\"b=20-2a\"\r\n" );
document.write( "\r\n" );
document.write( "Since the distance AB = 10\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%28%28a-0%29%5E2%2B%280-b%29%5E2%29=10\"\r\n" );
document.write( "\"sqrt%28a%5E2%2Bb%5E2%29=10\"\r\n" );
document.write( "\"a%5E2%2Bb%5E2=100\"\r\n" );
document.write( "\r\n" );
document.write( "\"a%5E2%2B%2820-2a%29%5E2=100\"\r\n" );
document.write( "\"a%5E2%2B%28400-40a%2B4a%5E2%29=100\"\r\n" );
document.write( "\"5a%5E2-40a=100\"\r\n" );
document.write( "\"5a%5E2-40a-100=0\"\r\n" );
document.write( "\"a%5E2-8a-20=0\"\r\n" );
document.write( "\"%28a%2B2%29%28a-20%29=0\"\r\n" );
document.write( "a+2=0;  a-20=0\r\n" );
document.write( "  a=-2;    a=20\r\n" );
document.write( "\"b=20-2a\"\r\n" );
document.write( "\"b=20-2%28-2%29\", \"b=20-2%2820%29\"\r\n" );
document.write( "\"b=20%2B4\",     \"b=20-20\"\r\n" );
document.write( "\"b=24\",       \"b=0\"\r\n" );
document.write( "\r\n" );
document.write( "So at first, there seem to be two answers. The first one\r\n" );
document.write( "\r\n" );
document.write( "a=-2, b=24\r\n" );
document.write( "\r\n" );
document.write( "\"x%2F%28-2%29+%2B+y%2F24+=+1\"\r\n" );
document.write( "\"-12x%2By=24\"\r\n" );
document.write( "\"y=12x%2B24\"\r\n" );
document.write( "with A=(a,0)=(-2,0), B=(0,24) \r\n" );
document.write( "and midpoint M of AB = \"%28matrix%281%2C3%2Ca%2F2%2C%22%2C%22%2Cb%2F2%29%29\" = \"%28matrix%281%2C3%2C%28-2%29%2F2%2C%22%2C%22%2C%28%2824%29%29%2F2%29%29\" = (-1,12)\r\n" );
document.write( "\r\n" );
document.write( "Here is solution with a=-2, b=24. The green line is 2x + y = 10.\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "However what seemed like another solution\r\n" );
document.write( "\r\n" );
document.write( "a=20, b=0\r\n" );
document.write( "\r\n" );
document.write( "turned out to be extraneous because when we substitute a=20, b=0\r\n" );
document.write( "in \"x%2Fa+%2B+y%2Fb+=+1\", we get\r\n" );
document.write( "\r\n" );
document.write( "\"x%2F20+%2B+cross%28y%2F0%29+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "We have an undefined term in the equation, since division by 0 is\r\n" );
document.write( "not permitted.\r\n" );
document.write( "\r\n" );
document.write( "If the equation had first been multiplied through by ab so that we\r\n" );
document.write( "would have had \"bx%2Bay=ab\" then we may have had a second solution, \r\n" );
document.write( "but since it was given as \"x%2Fa+%2B+y%2Fb+=+1\", the \"second solution\" \r\n" );
document.write( "was extraneous.  So there is just one solution a=-2, b=24.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );