document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=27828>Question 27828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hyperbolas,<BR>\n" );
document.write( "i need to write an equations for a hyperbola centered at (-4,5) and has a vertical transverse axis.  the value of \"a\" is 4 and the value of \"b\" is 9.<BR>\n" );
document.write( "         help me please!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15213 by </B><A HREF=/tutors/aboutme.mpl?userid=venugopalramana><B>venugopalramana(3286)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=venugopalramana><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.geocities.com/venugopalramana/FREE_MATHS_ASSISTANCE.html valign=middle><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=15213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> SEE FOLLOWING TO UNDERSTAND ASYMPTOTES AND THE AXES <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C%28%28%28%28%28x%5E2%29%2F9%29-1%29%2A9%29%5E0.5%29+%2C-%28%28%28%28%28x%5E2%29%2F9%29-1%29%2A9%29%5E0.5%29%2C%28%28%28%28%28x%5E2%29%2F9%29%2B1%29%2A9%29%5E0.5%29+%2C-%28%28%28%28%28x%5E2%29%2F9%29%2B1%29%2A9%29%5E0.5%29%2Cx%2C-x%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "YOU WILL FIND 2 SYMMETRIC CURVES LYING HORIZONTALLY..THEY ARE A PAIR OF HYPERBOLAS .THEIR EQN.IS <BR>\n" );
document.write( "(X^2/9)-(Y^2/9)=1...THE GENERAL EQN. IS (X^2/A^2)-(Y^2/B^2)=1<BR>\n" );
document.write( "THE OTHER PAIR OF HYPERBOLAS WHICH ARE IN A VERTICAL POSITION ARE CALLED CONJUGATE HYPERBOLAS OF THE EARLIER 2 HYPERBOLAS.THEIR EQN.IS <BR>\n" );
document.write( "(X^2/9)-(Y^2/9)= -1...THE GENERAL EQN. IS (X^2/A^2)-(Y^2/B^2)= -1<BR>\n" );
document.write( "NOW WE DEFINE THE VARIOUS TERMS WITH RESPECT TO THE FIRST HYPERBOLAS GIVEN BY<BR>\n" );
document.write( "(X^2/9)-(Y^2/9)=1...THE GENERAL EQN. IS (X^2/A^2)-(Y^2/B^2)=1<BR>\n" );
document.write( "DEFINITIONS.......<BR>\n" );
document.write( "HYPERBOLA IS THE LOCUS (OR PATH TRACED)BY A POINT WHICH MOVES SUCH THAT ITS DISTANCE FROM A FIXED POINT CALLED FOCUS TO ITS DISTANCE FROM A FIXED LINE CALLED DIRECTRIX IS CONSTANT (KNOWN AS ECCENTRICITY)AND IS MORE THAN 1.<BR>\n" );
document.write( "THERE ARE 2 FOCI AND 2 DIRECTIXES FOR THE HYPERBOLA GIVEN BY THE ABOVE DEFINITION AND EQN.<BR>\n" );
document.write( "X AXIS ALONG WHICH THE 2 HYPERBOLAS LIE IS CALLED TRANSVERSE AXIS.ITS EQN.IS Y=0<BR>\n" );
document.write( "Y AXIS ALONG WHICH THE 2 CONJUGATE HYPERBOLAS ARE PRESENT IS CALLED THE CONJUGATE AXIS.ITS EQN.IS X=0<BR>\n" );
document.write( "IF WE CALL THE 2 POINTS ON EITHER SIDE OF ORIGIN ON THE TRANSVERSE AXIS AT DISTANCE OF A FROM THE ORIGIN ARE NAMED A AND A' THEN AA'=2A IS THE LENGTH OF TRANVERSE AXIS<BR>\n" );
document.write( "IF WE CALL THE 2 POINTS ON EITHER SIDE OF ORIGIN ON THE CONJUGATE AXIS AT DISTANCE OF B FROM THE ORIGIN ARE NAMED B AND B' THEN BB'=2B IS THE LENGTH OF CONJUGATE AXIS<BR>\n" );
document.write( "ORIGIN IS THE CENTRE OF THE HYPERBOLAS<BR>\n" );
document.write( "ECCENTRICITY OF HYPERBOLA IS GIVEN BY E={(A^2+B^2)/(A^2)}^0.5<BR>\n" );
document.write( "FOCI ARE GIVEN BY (A/E,0)AND(-A/E,0)<BR>\n" );
document.write( "EQNS.OF DIRECTRIX 1 AND 2  ARE GIVEN BY X=A/E AND X=-A/E.<BR>\n" );
document.write( "THE 2 LINES YOU FIND DIAGONALLY ALMOST RUNNING PARALLEL TO THE CURVES AT THEIR ENDS ARE CALLED ASYMPTOTES.THE CURVES APPROACH THESE LINES AS NEAR AS WE DESIRE AT AS FAR A DISTANCE AS NEEDED,BUT NEVER TOUCH THEM .THEY RUN PARALLEL AS THE CURVES AND THE LINES EXTEND TO INFINITY .<BR>\n" );
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document.write( "GIVEN BELOW ARE SOME MORE EXAMPLES.NOW YOU TRY TO DRAW YOUR REQUIRED CURVE  AND UNDERSTAND.IF STILL IN DIFFICULTY COME BACK. <BR>\n" );
document.write( "IF CENTRE IS NOT ORIGIN BUT H,K ,THEN THE EQN OF HYPERBOLA WILL BE <BR>\n" );
document.write( "((X-H)^2/A^2)-((Y-K)^2/B^2)=1\r<BR>\n" );
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document.write( "Find the equations of the vertical and horizontal asymptotes for the graph of the rational function whose equation is f(x) = x/x+3.<BR>\n" );
document.write( "LET Y =X/(X+3)..D.R IS ZERO AT X=-3..SO THIS IS A CRITICAL POINT WHERE THE FUNCTION IS NOT DEFINED.<BR>\n" );
document.write( "HENCE WE SPLIT THE DOMAIN OF X<BR>\n" );
document.write( "1. FROM -INFINITY TO LESSTHAN  -3<BR>\n" );
document.write( "2. AND GREATER THAN   -3 TO +INFINITY<BR>\n" );
document.write( "THE GRAPH FOR DOMAIN <BR>\n" );
document.write( "1. FROM -INFINITY TO LESSTHAN  -3 IS AS FOLLOWS.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-50%2C1%2C+-1%2C+50%2C+x%2F%28x%2B3%29%2C1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-50%2C1%2C+-1%2C+50%2C+x%2F%28x%2B3%29%2C1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "ALGEBRAICALLY , WE FIND THE RANGE OF Y VARIES FROM 1 TO  INFINITY.<BR>\n" );
document.write( "SO ASYMPTOTES ARE Y=1 AS X TENDS TO MINUS INFINITY <BR>\n" );
document.write( "AND Y TENDING TO INFINITY AS X APPROACHES -3<BR>\n" );
document.write( "THE GRAPH FOR DOMAIN <BR>\n" );
document.write( "2. FROM GREATER THAN   -3 TO +INFINITY IS AS FOLLOW<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+500%2C+500%2C+-4%2C+50%2C+-50%2C+2%2C+x%2F%28x%2B3%29+%2C1%29+ ALIGN=MIDDLE  ALT=\"+graph%28+500%2C+500%2C+-4%2C+50%2C+-50%2C+2%2C+x%2F%28x%2B3%29+%2C1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "ALGEBRAICALLY , WE FIND THE RANGE OF Y VARIES FROM MINUS INFINITY  TO 1.<BR>\n" );
document.write( "SO ASYMPTOTES ARE Y=1 AS X TENDS TO INFINITY <BR>\n" );
document.write( "AND Y TENDING TO MINUS INFINITY AS X APPROACHES -3<BR>\n" );
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