document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085118>Question 1085118</A>: <I><SPAN STYLE=\"word-break: break-all;\"> P(x,y)is a variable point and A(2,2) is a fixed point. Find the equation of the locust of P if it moves so that AP is always equal to the perpendicular distance from the line y=3.Identity the locus and show it on the diagram.<BR>\n" );
document.write( "When is this locus of P less than zero? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #699223 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=699223\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"4\"><BR>\n" );
document.write( "Note: the term is locus (not locust)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Given points<BR>\n" );
document.write( "A = (2,2)<BR>\n" );
document.write( "P = (x,y)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Define point B as B = (x,3) where the 'x' is the same as the x coordinate of point P. This point B is directly above point P. Point B is located on the horizontal line y = 3.<BR>\n" );
document.write( "For a visual, see the diagram shown at the bottom of the page.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The distance from B to P is notated as d(B,P)<BR>\n" );
document.write( "The distance from A to P is notated as d(A,P)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In this case,<BR>\n" );
document.write( "d(B,P) = 3-y <BR>\n" );
document.write( "since this is the vertical distance from B to P\r<BR>\n" );
document.write( "\n" );
document.write( "Also,<BR>\n" );
document.write( "d(A,P) = sqrt((x-2)^2+(y-2)^2)<BR>\n" );
document.write( "using the distance formula\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equate the two distance expressions and solve for y<BR>\n" );
document.write( "d(B,P) = d(A,P)<BR>\n" );
document.write( "3-y = sqrt((x-2)^2+(y-2)^2)<BR>\n" );
document.write( "(3-y)^2 = (x-2)^2+(y-2)^2<BR>\n" );
document.write( "9 - 6y + y^2 = x^2 - 4x + 4 + y^2 - 4y + 4<BR>\n" );
document.write( "9 - 6y = x^2 - 4x + 4 - 4y + 4<BR>\n" );
document.write( "-2y = x^2 - 4x - 1<BR>\n" );
document.write( "y = (-1/2)*(x^2 - 4x - 1)<BR>\n" );
document.write( "y = (-1/2)*(x^2) + (-1/2)*(-4x) + (-1/2)*(-1)<BR>\n" );
document.write( "y = -0.5x^2 + 2x + 0.5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The equation is a parabola. The specific equation is <font color=red>y = -0.5x^2 + 2x + 0.5</font> where P is the point on the parabola (allowed to slide around) and point A is the focus (fixed). The line y = 3 is the directrix.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is what the graph would look like. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/kDak131.png\"><BR>\n" );
document.write( "Image generated by <a  rel=nofollow HREF= \"https://www.geogebra.org/home?ggbLang=en\">GeoGebra</a> (free graphing software).<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );