document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=823774>Question 823774</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A point P moves such that it is always twice as far from the points (4,0) as it is on the line x = 1\r<BR>\n" );
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document.write( "a) find the equation of the locus<BR>\n" );
document.write( "b) identify the locus and graph \r<BR>\n" );
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document.write( "Thanks so much in advance:)  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #495794 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=495794\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let (x, y) be the point P. Then the distance from point P to (4, 0) would be (using the distance formula):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%28x-4%29%5E2%2B%28y-0%29%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28x-4%29%5E2%2B%28y-0%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-8x%2B16%2By%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2-8x%2B16%2By%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The distance between a point and a line is always measured perpendicularly. Since the given line x = 1 is a vertical line, we will measure the distance from point P to the line x = 1 horizontally. A horizontal distance is simply the difference between the x-coordinates. The x-coordinates of all the points on the line x = 1 are 1's. Since we do not know if the x-coordinate of P is greater than or less than 1 we do not know which to put first in the difference. This problem can be solved using absolute value. This makes the distance from point P to the line x = 1:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-1%29 ALIGN=MIDDLE  ALT=\"abs%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "With the two expressions for the distances we can translate \"A point P moves such that it is always twice as far from the points (4,0) as it is on the line x = 1\" into the equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%5E2-8x%2B16%2By%5E2%29+=+2%2Aabs%28x-1%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%5E2-8x%2B16%2By%5E2%29+=+2%2Aabs%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we solve. Squaring both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16%2By%5E2+=+%282%2Aabs%28x-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2By%5E2+=+%282%2Aabs%28x-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16%2By%5E2+=+%282%29%5E2%2A%28abs%28x-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2By%5E2+=+%282%29%5E2%2A%28abs%28x-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Squaring the 2 is simple. But how do you square an absolute value? The answer may be obvious to you. If not then a basic property of absolute values can help:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28a%29%2Aabs%28b%29+=+abs%28a%2Ab%29 ALIGN=MIDDLE  ALT=\"abs%28a%29%2Aabs%28b%29+=+abs%28a%2Ab%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now, if we rewrite <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28abs%28x-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"%28abs%28x-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > without an exponent:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-1%29%2Aabs%28x-1%29 ALIGN=MIDDLE  ALT=\"abs%28x-1%29%2Aabs%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and then use the property we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%28x-1%29%28x-1%29%29 ALIGN=MIDDLE  ALT=\"abs%28%28x-1%29%28x-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%28x-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28%28x-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we have the absolute value of a perfect square. But perfect squares can never be negative. And since the absolute value of a non-negative is simply itself, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%28x-1%29%5E2%29 ALIGN=MIDDLE  ALT=\"abs%28%28x-1%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is just equal to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Back to squaring the right side of:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16%2By%5E2+=+%282%29%5E2%2A%28abs%28x-1%29%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2By%5E2+=+%282%29%5E2%2A%28abs%28x-1%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies as follows:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16%2By%5E2+=+4%2A%28x-1%29%5E2 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2By%5E2+=+4%2A%28x-1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16%2By%5E2+=+4%2A%28x%5E2-2x%2B1%29 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2By%5E2+=+4%2A%28x%5E2-2x%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2-8x%2B16%2By%5E2+=+4x%5E2-8x%2B4%29 ALIGN=MIDDLE  ALT=\"x%5E2-8x%2B16%2By%5E2+=+4x%5E2-8x%2B4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Last of all we put it in general standard form. For this we gather all the terms on one side. Subtracting the entire right side from both sides we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2B12%2By%5E2+=+0 ALIGN=MIDDLE  ALT=\"-3x%5E2%2B12%2By%5E2+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Reordering:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%5E2%2By%5E2%2B12+=+0 ALIGN=MIDDLE  ALT=\"-3x%5E2%2By%5E2%2B12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying both sides by -1 (because I prefer positive leading coefficients):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-y%5E2-12+=+0 ALIGN=MIDDLE  ALT=\"3x%5E2-y%5E2-12+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "At this point we should recognize the equation as the equation of a hyperbola. To help you graph it, let's put it in the standard form for hyperbolas:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2+-+%28y-k%29%5E2%2Fb%5E2+=+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2+-+%28y-k%29%5E2%2Fb%5E2+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding 12 to both sides of our equation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2-y%5E2=12 ALIGN=MIDDLE  ALT=\"3x%5E2-y%5E2=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing both sides by 12:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%5E2%2F12-y%5E2%2F12=1 ALIGN=MIDDLE  ALT=\"3x%5E2%2F12-y%5E2%2F12=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The first fraction reduces:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2F4-y%5E2%2F12=1 ALIGN=MIDDLE  ALT=\"x%5E2%2F4-y%5E2%2F12=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which can be rewritten as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-0%29%5E2%2F4-%28y-0%29%5E2%2F12=1 ALIGN=MIDDLE  ALT=\"%28x-0%29%5E2%2F4-%28y-0%29%5E2%2F12=1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "From this we can see that the center is the point (0, 0), the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+=+4 ALIGN=MIDDLE  ALT=\"a%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2+=+12 ALIGN=MIDDLE  ALT=\"b%5E2+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" >. I'll leave the graph up to you. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );