document.write( "Question 593105: Given A=(2,)) and B=(-1,0). Describe the set of points P satisfying the equation AP/PB=2 \n" ); document.write( "
Algebra.Com's Answer #376192 by jsmallt9(3758)\"\" \"About 
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Since \")\" is \"lower case\" 0, I'm going to assume that point A is (2, 0).

\n" ); document.write( "AP represents the length of segment AP (which is the distance between point A and point P. Using (x, y) for the point P and the distance formula we get:
\n" ); document.write( "AP = \"sqrt%28%28x-2%29%5E2+%2B+%28y-0%29%5E2%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "AP = \"sqrt%28x%5E2-4x%2B4+%2B+y%5E2%29\"

\n" ); document.write( "Using the distance formula in a similar way we can find PB:
\n" ); document.write( "PB = \"sqrt%28%28x-%28-1%29%29%5E2+%2B+%28y-0%29%5E2%29\"
\n" ); document.write( "Simplifying this we get:
\n" ); document.write( "PB = \"sqrt%28%28x%2B1%29%5E2+%2B+%28y-0%29%5E2%29\"
\n" ); document.write( "PB = \"sqrt%28x%5E2%2B2x%2B1+%2B+y%5E2%29\"
\n" ); document.write( "

\n" ); document.write( "Substituting these expressions into the given equation AP/PB = 2 we get:
\n" ); document.write( "\"sqrt%28x%5E2-4x%2B4+%2B+y%5E2%29%2Fsqrt%28x%5E2%2B2x%2B1+%2B+y%5E2%29+=+2\"

\n" ); document.write( "Next we will transform this equation into the general form for conic sections:
\n" ); document.write( "\"Ax%5E2+%2B+Bxy+%2B+Cy%5E2+%2BDx+%2B+Ey+%2B+F+=+0\"
\n" ); document.write( "We start by squaring both sides to eliminate the square roots:
\n" ); document.write( "\"%28x%5E2-4x%2B4+%2B+y%5E2%29%2F%28x%5E2%2B2x%2B1+%2B+y%5E2%29+=+4\"
\n" ); document.write( "Next we will multiply each side by the denominator (to get rid of the fraction):
\n" ); document.write( "
\n" ); document.write( "which gives us:
\n" ); document.write( "\"x%5E2-4x%2B4+%2B+y%5E2+=+4x%5E2%2B8x%2B4+%2B+4y%5E2\"
\n" ); document.write( "Now we just subtract \"x%5E2-4x%2B4+%2B+y%5E2\" from each side. (The general form is best written with a positive leading coefficient.) This gives us:
\n" ); document.write( "\"0+=+3x%5E2%2B12x+%2B+3y%5E2\"
\n" ); document.write( "Dividing each side by 3 we get:
\n" ); document.write( "\"0+=+x%5E2%2B+4x+%2B+y%5E2\"

\n" ); document.write( "With A = C one would think that this would be a circle. But let's complete the squares to make sure:
\n" ); document.write( "\"4+=+%28x%2B2%29%5E2+%2B+%28y-0%29%5E2\"
\n" ); document.write( "We can now see that the equation is indeed the equation of a circle (with a center at (-2, 0) and a radius of 2).

\n" ); document.write( "We needed to complete the squares because if we had gotten:
  • \"0+=+%28x%2B2%29%5E2+%2B+%28y-0%29%5E2\"
    \n" ); document.write( "we would have a \"circle of radius 0\" which is just a single point: (-2, 0).
  • any negative number = \"%28x%2B2%29%5E2+%2B+%28y-0%29%5E2\"
    \n" ); document.write( "we get nothing. This equation has NO solutions.
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