document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=28184>Question 28184</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the vertices, foci, and slope of the asymptotes for the hyperbola whose equation is, y^2/16 - x^2/25 =25? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #15970 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=15970\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> SEE THE FOLLOWING AND YOU SHOULD BE ABLE TO SOLVE YOUR PROBLEM BY YOUR SELF.....<BR>\n" );
document.write( "THE ANSWERS FOR YOUR CASE...H=0...K=0..A=4...B=5....EQN IS OF THE TYPE <BR>\n" );
document.write( "(Y-K)^2/B^2-(X-H)^2/A^2=1....<BR>\n" );
document.write( "SO VERTICES ARE...(H,(K-B)) AND  (H,(K+B)) ...(0,-5) AND (0,5)<BR>\n" );
document.write( "FOCI ARE {H,(K-BE)} AND {H,(K+BE)}...WHERE E IS <BR>\n" );
document.write( "ECCENTRICITY =SQRT{(A^2+B^2)/B^2}=SQRT((16+25)/25)=SQRT(41/25)<BR>\n" );
document.write( "SO FOCI ARE =(0,-5SQRT(41/25) AND (0,5SQRT(41/25)...<BR>\n" );
document.write( "OR....(0,-SQRT(41)) AND (0,SQRT(41)<BR>\n" );
document.write( "ASYMPTOTES ARE GIVEN BY <BR>\n" );
document.write( "Y^2/16-X^2/25=K<BR>\n" );
document.write( "25Y^2-16X^2-400K=0<BR>\n" );
document.write( "(5Y+4X+A)(5Y-4X+B)=0<BR>\n" );
document.write( "SLOPES OF ASYMPTOTES ARE <BR>\n" );
document.write( "-4/5 AND 4/5<BR>\n" );
document.write( "THE GRAPHS LOOK LIKE THIS<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+4%2A%281%2Bx%5E2%2F25%29%5E0.5%2C-4%2A%281%2Bx%5E2%2F25%29%5E0.5%2C4%2Ax%2F5%2C-4%2Ax%2F5+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------<BR>\n" );
document.write( "What is the vertices, foci, and slope of the asymptote for the hyperbola whose equation is, y^2 - 4x^2 - 2y - 16x + 1 = 0?<BR>\n" );
document.write( "(Y^2-2*Y*1+1^2)-{(2X)^2+2*(2X)*4+4^2}-1^2+4^2+1=0<BR>\n" );
document.write( "(Y-1)^2-(2X+4)^2=-16<BR>\n" );
document.write( "4(X+2)^2-(Y-1)^2=16<BR>\n" );
document.write( "{4(X+2)^2}/16-{(Y-1)^2}/16=1<BR>\n" );
document.write( "(X+2)^2/2^2-(Y-1)^2/4^2=1...<BR>\n" );
document.write( "COMPARING WITH STANDARD EQN. <BR>\n" );
document.write( "(X-H)^2/A^2-(Y-K)^2/B^2=1....WE HAVE <BR>\n" );
document.write( "VERTICES ARE {(H-A),K} AND {(H+A),K}=(-2-2,1) AND (-2+2,1)=(-4,1) AND (0,1)<BR>\n" );
document.write( "FOCI ARE {(H-AE),K} AND {(H+AE),K}...WHERE E IS <BR>\n" );
document.write( "ECCENTRICITY =SQRT{(A^2+B^2)/A^2}=SQRT((4+16)/4)=SQRT(5)<BR>\n" );
document.write( "SO FOCI ARE =(-2-2SQRT(5),1) AND (-2+2SQRT(5),1)<BR>\n" );
document.write( "SLOPE OF ASYMPTOTE IS GIVEN BY DIFFERENTIATION.HAVE YOU BEEN TAUGHT?PLEASE INFORM.I SHALL COME BACK ON HEARING FROM YOU.<BR>\n" );
document.write( "or you can take this proposition as proved formula<BR>\n" );
document.write( "the pair of asymptotes for a conic is given by the same equation as the conic except for the constant term which has to be found using the condition for the equation to represent a pair of straight lines.<BR>\n" );
document.write( "HENCE EQN OF ASYMPTOTES IS GIVEN BY <BR>\n" );
document.write( "y^2 - 4x^2 - 2y - 16x + K=0  , WHERE K IS DETERMINED using the condition for the equation to represent a pair of straight lines.<BR>\n" );
document.write( "SINCE WE ARE TO FIND ONLY SLOPES ,WE NEED NOT DETERMINE THE CONSTANT BUT ASSUME THAT THIS EQN REPRESENTS A PAIR OF STRAIGHT LINES.SO<BR>\n" );
document.write( "y^2 - 4x^2 - 2y - 16x + K=0 = (Y+2X+A)(Y-2X+B)<BR>\n" );
document.write( "HENCE SLOPES ARE +2 AND -2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+1-2%2A%28%28x%2B2%29%5E2-4%29%5E0.5%2C1%2B2%2A%28%28x%2B2%29%5E2-4%29%5E0.5%2C2x%2B5%2C-2x-3+%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );