document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1071489>Question 1071489</A>: <I><SPAN STYLE=\"word-break: break-all;\">  A system containing a circle and a parbola has 3 solutions. Find the system ( set of equations of the circle and the parbola) using the following information. Show your work.\r<BR>\n" );
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document.write( "&#12539;The center of the circle is at the origin (0,0)<BR>\n" );
document.write( "&#12539;The parbola opens upward.<BR>\n" );
document.write( "&#12539;The vertex of parbola is on the y=axis.<BR>\n" );
document.write( "&#12539;one of the solutions  is (root(7),3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #686428 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</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=686428\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A circle with radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%3E0 ALIGN=MIDDLE  ALT=\"R%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and center at the origin (0,0)<BR>\n" );
document.write( "has the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=R%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "A parabola, opening upwards, and with its vertex on the y-axis<BR>\n" );
document.write( "has the equation<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2%2Bb ALIGN=MIDDLE  ALT=\"y=ax%5E2%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with some constants <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3E0 ALIGN=MIDDLE  ALT=\"a%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Such a parabola has the y-axis as its axis of symmetry.<BR>\n" );
document.write( "You could also say the the y-axis is an axis of symmetry for<BR>\n" );
document.write( "the circle with radius <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R ALIGN=MIDDLE  ALT=\"R\"   DISABLED_style= \"max-width:90%; height:auto;\" > and center at the origin (0,0).<BR>\n" );
document.write( "As a consequence of this y-axis symmetry, for every solution of this system<BR>\n" );
document.write( "(intersection point of the parabola and circle),<BR>\n" );
document.write( "its reflection across the y-axis will also be a solution.<BR>\n" );
document.write( "The only way to get 3 solution is for the odd solution to be<BR>\n" );
document.write( "its own reflection across the y-axis image,<BR>\n" );
document.write( "meaning that the odd intersection point is on the y-axis.<BR>\n" );
document.write( "The circle crosses the y-axis at points (0,R) and (0,-R).<BR>\n" );
document.write( "One of those points must be a point of the parabola,<BR>\n" );
document.write( "specifically the point with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "parabola vertex <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2Cb%29 ALIGN=MIDDLE  ALT=\"V%280%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, it is either <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2C-R%29 ALIGN=MIDDLE  ALT=\"V%280%2C-R%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2CR%29 ALIGN=MIDDLE  ALT=\"V%280%2CR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but a parabola opening upwards with vertex <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2CR%29 ALIGN=MIDDLE  ALT=\"V%280%2CR%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%3ER ALIGN=MIDDLE  ALT=\"y%3ER\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all points other than the vertex,<BR>\n" );
document.write( "and would touch the circle only at that one point,<BR>\n" );
document.write( "meaning that the system would have only one solution.<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-R ALIGN=MIDDLE  ALT=\"b=-R\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2C-R%29 ALIGN=MIDDLE  ALT=\"V%280%2C-R%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the vertex,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2-R ALIGN=MIDDLE  ALT=\"y=ax%5E2-R\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the equation of the parabola.<BR>\n" );
document.write( "The point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28sqrt%287%29%2C3%29 ALIGN=MIDDLE  ALT=\"P%28sqrt%287%29%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a solution, so it is<BR>\n" );
document.write( "a point of the circle and a point of the parabola.<BR>\n" );
document.write( "Substituting into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2By%5E2=R%5E2 ALIGN=MIDDLE  ALT=\"x%5E2%2By%5E2=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28sqrt%287%29%29%5E2%2B3%5E2=R%5E2 ALIGN=MIDDLE  ALT=\"%28sqrt%287%29%29%5E2%2B3%5E2=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B9=R%5E2 ALIGN=MIDDLE  ALT=\"7%2B9=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16=R%5E2 ALIGN=MIDDLE  ALT=\"16=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The equation for the circle is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28x%5E2%2By%5E2=16%29 ALIGN=MIDDLE  ALT=\"highlight%28x%5E2%2By%5E2=16%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16=R%5E2 ALIGN=MIDDLE  ALT=\"16=R%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R=sqrt%2816%29 ALIGN=MIDDLE  ALT=\"R=sqrt%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28R=4%29 ALIGN=MIDDLE  ALT=\"highlight%28R=4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting that value,<BR>\n" );
document.write( "and the coordinates of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28sqrt%287%29%2C3%29 ALIGN=MIDDLE  ALT=\"P%28sqrt%287%29%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > into <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax%5E2-R ALIGN=MIDDLE  ALT=\"y=ax%5E2-R\"   DISABLED_style= \"max-width:90%; height:auto;\" > , we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=a%28sqrt%287%29%29%5E2-4 ALIGN=MIDDLE  ALT=\"3=a%28sqrt%287%29%29%5E2-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=7a-4 ALIGN=MIDDLE  ALT=\"3=7a-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B4=7a ALIGN=MIDDLE  ALT=\"3%2B4=7a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7=7a ALIGN=MIDDLE  ALT=\"7=7a\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F7=a ALIGN=MIDDLE  ALT=\"7%2F7=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=1%29 ALIGN=MIDDLE  ALT=\"highlight%28a=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and the equation for the parabola is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y=x%5E2-4%29 ALIGN=MIDDLE  ALT=\"highlight%28y=x%5E2-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( " <BR>\n" );
document.write( "So, the system the problem asks us to \"reverse-engineer\" for is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28system%28x%5E2%2By%5E2=16%2Cy=x%5E2-4%29%29 ALIGN=MIDDLE  ALT=\"highlight%28system%28x%5E2%2By%5E2=16%2Cy=x%5E2-4%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );