document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1152643>Question 1152643</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the value/s of U so that the graph of the equation of Ux2 + y^2 − 2Ux = 0 is a hyperbola.\r<BR>\n" );
document.write( "\n" );
document.write( "Afterwards, identify the distance/s between the foci. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854695 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5398)</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=854695\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I believe the equation was meant to be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ux%5E2+%2B+y%5E2+-+2Ux+=+0 ALIGN=MIDDLE  ALT=\"Ux%5E2+%2B+y%5E2+-+2Ux+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "FACTS ABOUT HYPERBOLAS:<BR>\n" );
document.write( "The equation for a hyperbola centered at (h,k) can be written as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=%22+%22%2B-+1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=%22+%22%2B-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some pair (a,b) of positive numbers.<BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=-1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fb%5E2-%28x-h%29%5E2%2Fa%5E2=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fb%5E2-%28x-h%29%5E2%2Fa%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fb%5E2=%28x-h%29%5E2%2Fa%5E2%2B1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fb%5E2=%28x-h%29%5E2%2Fa%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , you see that it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y-k%29%5E2%2Fb%5E2%3E=1 ALIGN=MIDDLE  ALT=\"%28y-k%29%5E2%2Fb%5E2%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28y-k%29%3E=b ALIGN=MIDDLE  ALT=\"abs%28y-k%29%3E=b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so the graph will have an upper branch and a lower branch, like this: <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C1%2C19%2C1%2C19%2Ccircle%2810%2C15%2C0.1%29%2Ccircle%2810%2C5%2C0.1%29%2C%0D%0Alocate%289.8%2C16%2CF%29%2Clocate%289.8%2C4.7%2C%22F%27%22%29%2Clocate%2812%2C9.95%2Ca%29%2C%0D%0Alocate%2810.2%2C12%2Cb%29%2Clocate%2810.2%2C9%2Cb%29%2Clocate%2812%2C11.7%2Cc%29%2C%0D%0Arectangle%286%2C7%2C10%2C13%29%2Crectangle%286%2C7%2C14%2C10%29%2Crectangle%286%2C7%2C14%2C13%29%2C%0D%0Aarc%2810%2C10%2C10%2C10%2C-150%2C-32%29%2Carc%2810%2C10%2C10%2C10%2C32%2C150%29%2C%0D%0Agraph%28300%2C300%2C1%2C19%2C1%2C19%2C2.5%2B0.75x%2C17.5-0.75x%2C10%2Bsqrt%289%2B0.5625%28x-10%29%5E2%29%2C%0D%0A10-sqrt%289%2B0.5625%28x-10%29%5E2%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The red and green lines are the asymptotes, with slopes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Fa ALIGN=MIDDLE  ALT=\"b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2Fa ALIGN=MIDDLE  ALT=\"-b%2Fa\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "In the other hand, for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2%2B1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so it must be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2%3E=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2%3E=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28x-h%29%3E=a ALIGN=MIDDLE  ALT=\"abs%28x-h%29%3E=a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so thee is a left branch and a tight branch to the graph, that looks like this: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C1%2C19%2C1%2C19%2C%0D%0Arectangle%286%2C7%2C10%2C13%29%2Crectangle%286%2C7%2C14%2C10%29%2Crectangle%286%2C7%2C14%2C13%29%2C%0D%0Agraph%28300%2C300%2C1%2C19%2C1%2C19%2C2.5%2B0.75x%2C17.5-0.75x%2C10%2Bsqrt%280.5625%28x-10%29%5E2-9%29%2C%0D%0A10-sqrt%280.5625%28x-10%29%5E2-9%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "SOLVING THE PROBLEM:<BR>\n" );
document.write( "Adding <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ux%5E2+%2B+y%5E2+-+2Ux+=+0 ALIGN=MIDDLE  ALT=\"Ux%5E2+%2B+y%5E2+-+2Ux+=+0\"   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=Ux%5E2-2Ux%2BU%2By%5E2=U ALIGN=MIDDLE  ALT=\"Ux%5E2-2Ux%2BU%2By%5E2=U\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U%28x%5E2-2x%2B1%29%2By%5E2=U ALIGN=MIDDLE  ALT=\"U%28x%5E2-2x%2B1%29%2By%5E2=U\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U%28x-1%29%5E2%2By%5E2=U ALIGN=MIDDLE  ALT=\"U%28x-1%29%5E2%2By%5E2=U\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28%28x-1%29%5E2%2By%5E2%2FU=1%29 ALIGN=MIDDLE  ALT=\"highlight%28%28x-1%29%5E2%2By%5E2%2FU=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That equation represents a hyperbola for any negative value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "If allowed to choose a value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > to calculate distances, I would choose <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U=-1 ALIGN=MIDDLE  ALT=\"U=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2-y%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2-y%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > , matching the general equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-h%29%5E2%2Fa%5E2-%28y-k%29%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28h=1%2Ck=0%2Ca=1%2Cb=1%29 ALIGN=MIDDLE  ALT=\"system%28h=1%2Ck=0%2Ca=1%2Cb=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "The equation represents a hyperbola with center (1,0),<BR>\n" );
document.write( "vertices <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2C0%29 ALIGN=MIDDLE  ALT=\"V%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%282%2C0%29 ALIGN=MIDDLE  ALT=\"W%282%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , asymptotes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%22+%22%2B-+%28x-1%29 ALIGN=MIDDLE  ALT=\"y=%22+%22%2B-+%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , focal distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=sqrt%28a%5E2%2Bb%5E2%29=sqrt%282%29 ALIGN=MIDDLE  ALT=\"c=sqrt%28a%5E2%2Bb%5E2%29=sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and foci <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F ALIGN=MIDDLE  ALT=\"F\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22F%27%22 ALIGN=MIDDLE  ALT=\"%22F%27%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B-+sqrt%282%29 ALIGN=MIDDLE  ALT=\"1+%2B-+sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C+0%29%22 ALIGN=MIDDLE  ALT=\"%22%2C+0%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The distance between the foci is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%282%29 ALIGN=MIDDLE  ALT=\"2sqrt%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-2%2C4%2C-3%2C3%2Crectangle%280%2C-1%2C2%2C1%29%2C%0D%0Agraph%28400%2C400%2C-2%2C4%2C-3%2C3%2Cx-1%2C1-x%2Csqrt%28%28x-1%29%5E2-1%29%2C-sqrt%28%28x-1%29%5E2-1%29%29%2C%0D%0Acircle%280%2C0%2C0.05%29%2Ccircle%282%2C0%2C0.05%29%2Clocate%280.05%2C0.25%2CV%29%2Clocate%281.85%2C0.25%2CW%29%2C%0D%0Acircle%281%2Bsqrt%282%29%2C0%2C0.05%29%2Ccircle%281-sqrt%282%29%2C0%2C0.05%29%2C%0D%0Alocate%282%2C2.2%2Cred%28y=x-1%29%29%2Clocate%282%2C-2%2Cgreen%28y=-x%2B1%29%29%2C%0D%0Alocate%28-0.5%2C-0.07%2CF%29%2Clocate%282.35%2C-0.07%2C%22F%27%22%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If a generic <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U ALIGN=MIDDLE  ALT=\"U\"   DISABLED_style= \"max-width:90%; height:auto;\" > is expected, we can say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=U=-b%5E2 ALIGN=MIDDLE  ALT=\"U=-b%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3E0 ALIGN=MIDDLE  ALT=\"b%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5E2-y%5E2%2Fb%5E2=1 ALIGN=MIDDLE  ALT=\"%28x-1%29%5E2-y%5E2%2Fb%5E2=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That equation represents a hyperbola with center (1,0),<BR>\n" );
document.write( "vertices <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=V%280%2C0%29 ALIGN=MIDDLE  ALT=\"V%280%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=W%282%2C0%29 ALIGN=MIDDLE  ALT=\"W%282%2C0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , asymptotes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%22+%22%2B-+b%28x-1%29 ALIGN=MIDDLE  ALT=\"y=%22+%22%2B-+b%28x-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , focal distance <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=sqrt%281%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"c=sqrt%281%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and foci at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%28%22 ALIGN=MIDDLE  ALT=\"%22%28%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B-+sqrt%281%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"1+%2B-+sqrt%281%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22%2C+0%29%22 ALIGN=MIDDLE  ALT=\"%22%2C+0%29%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The distance between the foci is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2sqrt%281%2Bb%5E2%29 ALIGN=MIDDLE  ALT=\"2sqrt%281%2Bb%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );