document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1146375>Question 1146375</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Given a circle of radius of 4 and center at (6, 8). Find the locus of the center of all circles that is tangent to the x  -  axis and the given circle.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #767644 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=767644\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The required conditions for each point in the locus are<br><BR>\n" );
document.write( "(1) The radius of the circle is the y value; and<BR>\n" );
document.write( "(2) The radius of the circle plus 4 is the distance between the point and the center of the circle.<br><BR>\n" );
document.write( "Those two requirements give us, for any point (x,y) on the locus,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B4+=+sqrt%28%28x-6%29%5E2%2B%28y-8%29%5E2%29%29 ALIGN=MIDDLE  ALT=\"y%2B4+=+sqrt%28%28x-6%29%5E2%2B%28y-8%29%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2%2B8y%2B16+=+x%5E2-12x%2B36%2By%5E2-16y%2B64 ALIGN=MIDDLE  ALT=\"y%5E2%2B8y%2B16+=+x%5E2-12x%2B36%2By%5E2-16y%2B64\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+x%5E2-12x-24y%2B84 ALIGN=MIDDLE  ALT=\"0+=+x%5E2-12x-24y%2B84\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24y+=+%28x%5E2-12x%2B36%29%2B48 ALIGN=MIDDLE  ALT=\"24y+=+%28x%5E2-12x%2B36%29%2B48\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F24%29%28x-6%29%5E2%2B2 ALIGN=MIDDLE  ALT=\"y+=+%281%2F24%29%28x-6%29%5E2%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Here is another path to the same solution.<br><BR>\n" );
document.write( "With the circle with center (6,8) and radius 4, the point on the circle closest to the x-axis is (6,4); then clearly one point on the locus is (6,2) -- halfway between (6,4) and the x-axis.<br><BR>\n" );
document.write( "It should also by clear by symmetry that the locus will be symmetrical about the line x=6; any point on the locus \"a\" units to the right of the line x=6 will be mirrored by a point on the locus \"a\" units to the left of the line x=6.<br><BR>\n" );
document.write( "So the locus is symmetrical about the line x=6; it is therefore a parabola with vertex (6,2) and an equation of the form<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2+=+a%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y-2+=+a%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To determine the value of the constant a, note that a point on the locus to the right of x=6 will have a y value of 8 and an x value of 6 (center of circle) plus 4 (radius of circle) plus 8 (equal to the y value).<br><BR>\n" );
document.write( "That gives us the point (18,8).  Plugging those values in the equation of the parabola gives us the value of a:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8-2+=+a%2818-6%29%5E2 ALIGN=MIDDLE  ALT=\"8-2+=+a%2818-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+=+144a ALIGN=MIDDLE  ALT=\"6+=+144a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1%2F24 ALIGN=MIDDLE  ALT=\"a+=+1%2F24\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And again we find the equation of the locus i<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-2+=+%281%2F24%29%28x-6%29%5E2 ALIGN=MIDDLE  ALT=\"y-2+=+%281%2F24%29%28x-6%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "or<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%281%2F24%29%28x-6%29%5E2%2B2 ALIGN=MIDDLE  ALT=\"y+=+%281%2F24%29%28x-6%29%5E2%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );