document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1134394>Question 1134394</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A point P(2p,p^2) lie on x^2=4ay. The perpendicular, from the focus S of the parabola, on to the tangent cuts the directrix at M. If N is the midpoint of the interval PM, find the equation of the locus of N.\r<BR>\n" );
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document.write( "I keep getting long equations that I don't know how to simplify. Right now I have this: 2y+4ay+2a^y=x^2-a-2a^2-a^3 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #751969 by </B><A HREF=/tutors/aboutme.mpl?userid=t0hierry><B>t0hierry(194)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=t0hierry><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/cgi-bin/show-face.mpl?user=t0hierry><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=751969\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I don't know how to solve your problem. All I can do is write down for you the equation of a parabola.\r<BR>\n" );
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document.write( "There are two kinds, vertical and horizontal. IF the parabola is vertical, along y, then it is of the form {{y = Ax^2 + B}}. \r<BR>\n" );
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document.write( "You derive this by recalling that the distance to the focus is equal to the distance to the directrix.\r<BR>\n" );
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document.write( "{{(x - a)^2 + (y-b)^2 = (y-k)^2}}\r<BR>\n" );
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document.write( "Here the focus is a point of coordinates (a,b). And the directrix has equation y = k.\r<BR>\n" );
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document.write( "I think the equation you're looking for is similar to this one, which is why I go through its steps:\r<BR>\n" );
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document.write( "{{y^2 - 2ky + k^2 = y^2 -2by + b^2 + (x-a)^2}}\r<BR>\n" );
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document.write( "y^2 cancels out, as expected, and we end up with\r<BR>\n" );
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document.write( "{{y(2b - 2k) = b^2 - k^2 + (x-a)^2}} or<BR>\n" );
document.write( "{{y = (x-a)^2/[2(b-k)] + (b+k)/2}}\r<BR>\n" );
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document.write( "The other equation for the parabola, has a physical explanation. It's the trajectory of a point mass, dropped from a plane, or shot through a canon. There is no acceleration in the horizontal direction, so v = constant. <BR>\n" );
document.write( "And thus x = vt.<BR>\n" );
document.write( "At the same time, there is acceleration along the y direction, that of gravity. And naturally you have v = gt and {{y = 1/2 g t^2 =1/2 g(x/v)^2 = 1/2 g/v^2 x^2}}\r<BR>\n" );
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document.write( "I was hoping someone else solved this problem for you but, if they did, I missed it. Anyway, I hope this helps. </SPAN>\n" );
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