document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196068>Question 1196068</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph the line that is described parametrically by (x,y) = (2t, 5-t), then:\r<BR>\n" );
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document.write( "(a) Confirm that the point corresponding to t=0 is exactly 5 units from (3,9);<BR>\n" );
document.write( "(b) Write a formula in terms of t for the distance from (3,9) to (2t, 5-t)<BR>\n" );
document.write( "(c) Find one other point on the line that is 5 units from (3,9)<BR>\n" );
document.write( "(d) Find the point on the line that minimizes the distance to (3,9) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828766 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828766\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Given x=2t, find y in terms of x:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5-t=5-%281%2F2%29%282t%29=5-%281%2F2%29x ALIGN=MIDDLE  ALT=\"y=5-t=5-%281%2F2%29%282t%29=5-%281%2F2%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The equation of the line in slope-intercept form is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F2%29x%2B5 ALIGN=MIDDLE  ALT=\"y=%28-1%2F2%29x%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C200%2C-2%2C12%2C-1%2C6%2C%28-1%2F2%29x%2B5%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C200%2C-2%2C12%2C-1%2C6%2C%28-1%2F2%29x%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(a) The point corresponding to t=0 is (0,5).  From that point to (3,9) the difference in x is 3 and the difference in y is 4, so the distance to (3,9) is 5 (by the 3-4-5 Pythagorean Triple).<br><BR>\n" );
document.write( "(b) To avoid the confusion of the square root, I will find the formula for the square of the distance from (3,9) to a point on the line.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2=%283-2t%29%5E2%2B%289-%285-t%29%29%5E2=4t%5E2-12t%2B9%2Bt%5E2%2B8t%2B16=5t%5E2-4t%2B25 ALIGN=MIDDLE  ALT=\"d%5E2=%283-2t%29%5E2%2B%289-%285-t%29%29%5E2=4t%5E2-12t%2B9%2Bt%5E2%2B8t%2B16=5t%5E2-4t%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(c) Find where the square of the distance is 5^2=25:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5t%5E2-4t%2B25=25 ALIGN=MIDDLE  ALT=\"5t%5E2-4t%2B25=25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5t%5E2-4t=0 ALIGN=MIDDLE  ALT=\"5t%5E2-4t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%285t-4%29=0 ALIGN=MIDDLE  ALT=\"t%285t-4%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=4%2F5 ALIGN=MIDDLE  ALT=\"t=4%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "We already knew t=0 was one solution; t=4/5 is the other.  The point on the line corresponding to t=4/5 is (2t,5-t) = (8/5,21/5).<br><BR>\n" );
document.write( "CHECK: (3-8/5)^2+(9-21/5)^2 = (7/5)^2+(24/5)^2 = 49/25+576/25 = 625/25 = 25<br><BR>\n" );
document.write( "(d) By symmetry, the point on the line that minimizes the distance from (3,9) is halfway between the two points that are the same distance 5 from the line.  That corresponds to t halfway between 0 and 4/5, or t = 2/5.  That point on the line is (2t,5-t) = (4/5,23/5).<br><BR>\n" );
document.write( " </SPAN>\n" );
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