document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039424>Question 1039424</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the parabola y = x^2\r<BR>\n" );
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document.write( "a. find the equation of the tangent to the parabola at the point (t, t^2)<BR>\n" );
document.write( "b. show that the line passing through the focus of the parabola and perpendicular to the tangent in (a) has the equation y = t-2x/4t<BR>\n" );
document.write( "c. show that the foot of the perpendicular from the locus to the tangent is the point F(t/2 , 0)<BR>\n" );
document.write( "d. find the locus of M, the midpoint of PF\r<BR>\n" );
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document.write( "NEED HELP WITH PART C AND D!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654207 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=654207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "If the foot of the perpendicular to the tangent through the focus is indeed the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{t}{2},\,0\right)\">, then the zero of the function representing the tangent (i.e. the answer to part a) and the zero of the function representing the perpendicular to the tangent through the focus (i.e. the function given in part b) must both be equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{t}{2}\">.  I'll let you do the algebra to verify that the zeros are both <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{t}{2}\">.\r<BR>\n" );
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document.write( "I can't be sure about part d because you don't specify point P.  However, I'll go out on a limb and assume it is the focus.  In that case, just use the midpoint formulae with the two points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(0,\,\frac{1}{4}\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{t}{2},\,0\right)\">.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_m\ =\ \frac{x_1\ +\ x_2}{2}\ =\ \frac{0\ +\ \frac{t}{2}}{2}\ =\ \frac{t}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_m\ =\ \frac{y_1\ +\ y_2}{2}\ =\ \frac{\frac{1}{4}\ +\ 0}{2}\ =\ \frac{1}{8}\">\r<BR>\n" );
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document.write( "So for any given value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\">, M is the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(\frac{t}{4},\,\frac{1}{8}\right)\">.  Hence, for all real values of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> the locus of points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large M_t\"> is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ \frac{1}{8}\">\r<BR>\n" );
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document.write( "Assuming, of course, that P represents the focus.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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