document.write( "Question 421278: a boat travels so that it is twice as far from one buoy as another buoy 3 miles away. describe the path of the boat \n" ); document.write( "
Algebra.Com's Answer #294228 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "Let's place the buoys at the points (-1.5,0) and (1.5,0)\r\n" );
document.write( "so they will be 3 units (miles) apart:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let's draw an arbitrary variable point where the boat might be\r\n" );
document.write( "at one instant and label it (x,y) that appears to be about twice \r\n" );
document.write( "as far from (-1.5,0) as it is from (1.5,0), and draw a line from \r\n" );
document.write( "this arbitrary variable point to each of the buoy points (-1.5,0)\r\n" );
document.write( "and (1.5,0). Label the longer line d1 and the shorter \r\n" );
document.write( "one d2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We set d1 equal to 2 times d2\r\n" );
document.write( "\r\n" );
document.write( "\"d%5B1%5D=2d%5B2%5D\"\r\n" );
document.write( "\r\n" );
document.write( "Now we use the distance formula to substitute for the\r\n" );
document.write( "two distances d1 and d2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%28%28x%2B1.5%29%5E2%2B%28y-0%29%5E2%29+=+2sqrt%28%28x-1.5%29%5E2%2B%28y-0%29%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"sqrt%28%28x%2B1.5%29%5E2%2By%5E2%29+=+2sqrt%28%28x-1.5%29%5E2%2By%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Square both sides:\r\n" );
document.write( "\r\n" );
document.write( "\"%28sqrt%28%28x%2B1.5%29%5E2%2By%5E2%29%29%5E2+=+%282sqrt%28%28x-1.5%29%5E2%2By%5E2%29%29%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B1.5%29%5E2%2By%5E2+=+2%5E2%28sqrt%28%28x-1.5%29%5E2%2By%5E2%29%29%5E2\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B1.5%29%5E2%2By%5E2+=+4%28%28x-1.5%29%5E2%2By%5E2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x%2B1.5%29%28x%2B1.5%29%2By%5E2+=+4%28%28x-1.5%29%28x-1.5%29%2By%5E2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2%2B1.5x%2B1.5x%2B2.25%2By%5E2+=+4%28x%5E2-1.5x-1.5x%2B2.25%2By%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2%2B3x%2B2.25%2By%5E2+=+4%28x%5E2-3x%2B2.25%2By%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2%2B3x%2B2.25%2By%5E2+=+4x%5E2-12x%2B9%2B4y%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"0+=+3x%5E2-15x%2B6.75%2B3y%5E2%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"3x%5E2-15x%2B6.75%2B3y%5E2=0%29\"\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 3\r\n" );
document.write( "\r\n" );
document.write( "x² - 5x + 2.25 + y² = 0\r\n" );
document.write( "\r\n" );
document.write( "We can tell this is a circle.  We need to get it in\r\n" );
document.write( "the standard form (x-h)² + (y-k)² = r²\r\n" );
document.write( "Get the constant on the right:\r\n" );
document.write( "\r\n" );
document.write( "x² - 5x + y² = -2.25\r\n" );
document.write( "\r\n" );
document.write( "Complete the square on the x-terms.\r\n" );
document.write( "\r\n" );
document.write( "Multiply the coefficient of x, which is -5 by \"1%2F2\", \r\n" );
document.write( "getting -2.5.  Square 2.5, getting 6.25\r\n" );
document.write( "\r\n" );
document.write( "Add that to both sides:\r\n" );
document.write( "\r\n" );
document.write( "x² - 5x + y² = -2.25\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "x² - 5x + 6.25 + y² = -2.25 + 6.25\r\n" );
document.write( "\r\n" );
document.write( "(x - 2.5)(x - 2.5) + y² = 4\r\n" );
document.write( "\r\n" );
document.write( "Write the (x - 25)(x - 2.5) as (x - 2.5)² \r\n" );
document.write( "write y² as (y - 0)²\r\n" );
document.write( "Write 4 as 2²\r\n" );
document.write( "\r\n" );
document.write( "(x - 2.5)² + (y - 0)² = 2²\r\n" );
document.write( "\r\n" );
document.write( "So the boat travels in a circle with center (2.5,0)\r\n" );
document.write( "and radius 2 \r\n" );
document.write( "\r\n" );
document.write( "Here is the circle the boat travels in:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );