document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1157992>Question 1157992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The reflection property of parabolas. Consider the parabola whose focus is F = (1,4) and whose directrix is the line x = −3.<BR>\n" );
document.write( "(a) Sketch the parabola, and make calculations that confirm that P = (7, 12) is on it.<BR>\n" );
document.write( "(b) Find the slope of the line μ through P that is tangent to the parabola.<BR>\n" );
document.write( "(c) Calculate the size of the angle that μ makes with the line y = 12.<BR>\n" );
document.write( "(d) Calculate the size of the angle that μ makes with segment F P .  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854535 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5377)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> (a) Sketch the parabola, and make calculations that confirm that P = (7, 12) is on it.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C16%2C-6%2C14%2C%0D%0Agraph%28300%2C300%2C-4%2C16%2C-6%2C14%2Csqrt%288x%2B8%29%2B4%2C4-sqrt%288x%2B8%29%29%2C%0D%0Ablue%28triangle%28-3%2C-6%2C-3%2C14%2C-3%2C4%29%29%2Ccircle%287%2C12%2C0.2%29%2C%0D%0Alocate%282%2C-4.5%2Cblue%28directrix%29%29%2Clocate%287.1%2C12%2CP%287%2C12%29%29%2C%0D%0Ablue%28arrow%281.9%2C-5%2C-3%2C-5%29%29%2Ccircle%281%2C4%2C0.2%29%2C%0D%0Alocate%281.1%2C4%2CF%281%2C4%29%29%2Clocate%285%2C5%2Cparabola%29%2C%0D%0Ared%28arrow%285.6%2C5.1%2C5.6%2C11.27%29%29%2C%0D%0Agreen%28arrow%285.6%2C4.1%2C5.6%2C-3.27%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A parabola is the locus of the points that are at the same distance from the directrix and the focus.<BR>\n" );
document.write( "The distance between directrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-3 ALIGN=MIDDLE  ALT=\"x=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%287%2C12%29 ALIGN=MIDDLE  ALT=\"P%287%2C12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7-%28-3%29=7%2B3=10 ALIGN=MIDDLE  ALT=\"7-%28-3%29=7%2B3=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The distance between focus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%281%2C4%29 ALIGN=MIDDLE  ALT=\"F%281%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%287%2C12%29 ALIGN=MIDDLE  ALT=\"P%287%2C12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28%287-1%29%5E2%2B%2812-4%29%5E2%29=sqrt%286%5E2%2B8%5E2%29=sqrt%2836%2B64%29=sqrt%28100%29=10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%287%2C12%29 ALIGN=MIDDLE  ALT=\"P%287%2C12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is at the same distance from the directrix and the focus,<BR>\n" );
document.write( "so it is on the parabola.<BR>\n" );
document.write( "We could prove P is on the parabola from the equation of the parabola.<BR>\n" );
document.write( "Knowing that the equation of a parabola with its vertex at the origin and focal distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f ALIGN=MIDDLE  ALT=\"f\"   DISABLED_style= \"max-width:90%; height:auto;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F4f%29y%5E2 ALIGN=MIDDLE  ALT=\"x=%281%2F4f%29y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for parabolas with the x-axis as an axis of symmetry,<BR>\n" );
document.write( "we can translate that equation for a parabola with the vertex at (-1,4), halfway between focus and directrix.<BR>\n" );
document.write( "Doing that, we found that our parabola has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f=1-%28-1%29=2 ALIGN=MIDDLE  ALT=\"f=1-%28-1%29=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the equation for our parabola would be  <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-%28-1%29=%281%2F%284%2A2%29%29%28y-4%29%5E2 ALIGN=MIDDLE  ALT=\"x-%28-1%29=%281%2F%284%2A2%29%29%28y-4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=%281%2F8%29%28y-4%29%5E2 ALIGN=MIDDLE  ALT=\"x%2B1=%281%2F8%29%28y-4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12 ALIGN=MIDDLE  ALT=\"y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=%281%2F8%29%2812-4%29%5E2=%281%2F8%29%2A8%5E2=8 ALIGN=MIDDLE  ALT=\"x%2B1=%281%2F8%29%2812-4%29%5E2=%281%2F8%29%2A8%5E2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=8-1=7 ALIGN=MIDDLE  ALT=\"x=8-1=7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "(b) Find the slope of the line μ through P that is tangent to the parabola.<BR>\n" );
document.write( "We can estimate the slope of the tangent line from the graph or calculate it from the derivative of the function.<BR>\n" );
document.write( "From <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=%281%2F8%29%28y-4%29%5E2 ALIGN=MIDDLE  ALT=\"x%2B1=%281%2F8%29%28y-4%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%281%2F8%29%28y-4%29%5E2-1 ALIGN=MIDDLE  ALT=\"x=%281%2F8%29%28y-4%29%5E2-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can find the derivative<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdy=%281%2F8%29%2A2%28y-4%29 ALIGN=MIDDLE  ALT=\"dx%2Fdy=%281%2F8%29%2A2%28y-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%287%2C12%29 ALIGN=MIDDLE  ALT=\"P%287%2C12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , that derivative is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdy=%281%2F8%29%2A2%2812-4%29=%281%2F8%29%2A2%2A8=2 ALIGN=MIDDLE  ALT=\"dx%2Fdy=%281%2F8%29%2A2%2812-4%29=%281%2F8%29%2A2%2A8=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the function graphed in red above, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx=highlight%281%2F2%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx=highlight%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the derivative and slope of the tangent at point P.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(c) Calculate the size of the angle that μ makes with the line y = 12.<BR>\n" );
document.write( "(d) Calculate the size of the angle that μ makes with segment F P . <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-4%2C16%2C-2%2C18%2C%0D%0Agraph%28300%2C300%2C-4%2C16%2C-2%2C18%2Csqrt%288x%2B8%29%2B4%2C4-sqrt%288x%2B8%29%2Cx%2F2%2B8.5%29%2C%0D%0Acircle%287%2C12%2C0.2%29%2Clocate%28-3.9%2C6.5%2C+blue%28mu%29%29%2C%0D%0Alocate%287.1%2C12%2CP%287%2C12%29%29%2Ctriangle%281%2C4%2C7%2C12%2C4%2C8%29%2C%0D%0Acircle%281%2C4%2C0.2%29%2Ctriangle%28-4%2C12%2C7%2C12%2C2%2C12%29%2Ctriangle%28-12%2C12%2C7%2C12%2C16%2C12%29%2C%0D%0Alocate%281.1%2C4%2CF%281%2C4%29%29%2Clocate%28-3.9%2C12%2Cy=12%29%2C%0D%0Agreen%28arc%287%2C12%2C16%2C16%2C-26.6%2C0%29%29%2Clocate%2813%2C14%2Cgreen%28alpha%29%29%2C%0D%0Agreen%28arc%287%2C12%2C13%2C13%2C153.4%2C180%29%29%2Clocate%281%2C11%2Cgreen%28alpha%29%29%2C%0D%0Agreen%28arc%287%2C12%2C7%2C7%2C126.9%2C180%29%29%2Clocate%284.5%2C11%2Cgreen%28beta%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12 ALIGN=MIDDLE  ALT=\"y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > slope is zero, it is parallel to the x-axis.<BR>\n" );
document.write( "Line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%28mu%29 ALIGN=MIDDLE  ALT=\"blue%28mu%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > makes an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28alpha%29 ALIGN=MIDDLE  ALT=\"green%28alpha%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the x-axis and with the line<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12 ALIGN=MIDDLE  ALT=\"y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28green%28alpha%29%29=1%2F2 ALIGN=MIDDLE  ALT=\"tan%28green%28alpha%29%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28alpha%29=highlight%2826.565%5Eo%29 ALIGN=MIDDLE  ALT=\"green%28alpha%29=highlight%2826.565%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Segment FP, connecting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%281%2C4%29 ALIGN=MIDDLE  ALT=\"F%281%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >} and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%287%2C12%29 ALIGN=MIDDLE  ALT=\"P%287%2C12%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and line FP, have a slope of<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812-4%29%2F%287-1%29=8%2F6=4%2F3 ALIGN=MIDDLE  ALT=\"%2812-4%29%2F%287-1%29=8%2F6=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > . A line, or segment with such a slope would make an angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28beta%29 ALIGN=MIDDLE  ALT=\"green%28beta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with the x-axis and line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12 ALIGN=MIDDLE  ALT=\"y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=tan%28green%28beta%29%29=4%2F3 ALIGN=MIDDLE  ALT=\"tan%28green%28beta%29%29=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28beta%29=53.130%5Eo ALIGN=MIDDLE  ALT=\"green%28beta%29=53.130%5Eo\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The angle that μ makes with segment FP is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%28beta%29-green%28alpha%29=53.130%5Eo-26.565%5Eo=highlight%2826.565%5Eo%29 ALIGN=MIDDLE  ALT=\"green%28beta%29-green%28alpha%29=53.130%5Eo-26.565%5Eo=highlight%2826.565%5Eo%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );