document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=482730>Question 482730</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of a hyperbola whose asymptote coincide withe the points (-7, 4) and (-1, -14). Its other asymptote contains point (-4,1), and (-3/4, -53/4) is a point on the hyperbola. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #330591 by </B><A HREF=/tutors/aboutme.mpl?userid=lwsshak3><B>lwsshak3(11628)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=lwsshak3><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=330591\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Find the equation of a hyperbola whose asymptote coincide with the points (-7, 4) and (-1, -14). Its other asymptote contains point (-4,1), and (-3/4, -53/4) is a point on the hyperbola.<BR>\n" );
document.write( "**<BR>\n" );
document.write( "I will assume the given point -53/4 was meant to be written as (-5,3/4); otherwise, it didn't seem to be correct.<BR>\n" );
document.write( "..<BR>\n" );
document.write( "Asymptotes are straight lines of the standard form: y=mx+b, m=slope, b=y-intercept<BR>\n" );
document.write( "For asymptote with points (-7,4) and (-1,-14):<BR>\n" );
document.write( "slope, m =&#8710;y/&#8710;x=(-14-4)/(-1,-(-7))=-18/6=-3<BR>\n" );
document.write( "Equation: <BR>\n" );
document.write( "y=-3x+b<BR>\n" );
document.write( "solving for b using one of given points, (-7,4)<BR>\n" );
document.write( "4=-3*-7+b<BR>\n" );
document.write( "4=21+b<BR>\n" );
document.write( "b=-17<BR>\n" );
document.write( "Equation of asymptote:<BR>\n" );
document.write( "y=-3x-17<BR>\n" );
document.write( "..<BR>\n" );
document.write( "For asymptote with points (-4,1) and (-3,4):<BR>\n" );
document.write( "slope, m =&#8710;y/&#8710;x=(4-1)/(-3,-(-4)=3/1=3<BR>\n" );
document.write( "Equation: <BR>\n" );
document.write( "y=3x+b<BR>\n" );
document.write( "solving for b using one of given points, (-4,1)<BR>\n" );
document.write( "1=3*(-4)+b<BR>\n" );
document.write( "1=-12+b<BR>\n" );
document.write( "b=13<BR>\n" );
document.write( "Equation of asymptote:<BR>\n" );
document.write( "y=3x+13<BR>\n" );
document.write( "..<BR>\n" );
document.write( "Point of intersection of asymptotes=center of hyperbola<BR>\n" );
document.write( "-3x-17=3x+13<BR>\n" );
document.write( "6x=-30<BR>\n" );
document.write( "x=-5<BR>\n" );
document.write( "y=3x+13=-15+13=-2<BR>\n" );
document.write( "Center of hyperbola: (-5,-2)<BR>\n" );
document.write( "..<BR>\n" );
document.write( "Note that the given point on the hyperbola(-5,3/4) is on the same line as the center (-5,2) which means the hyperbola has a vertical transverse axis of the standard form: (y-k)^/a^2-(x-h)^2/b^2=1.<BR>\n" );
document.write( "The point (-5,3/4) are also the coordinates of the (upper) vertex.<BR>\n" );
document.write( "..<BR>\n" );
document.write( "a=half the length of the vertical transverse axis or the distance from the center to the vertex=2+3/4=11/4<BR>\n" );
document.write( "a=11/4<BR>\n" );
document.write( "slope =3=a/b<BR>\n" );
document.write( "b=a/3=(11/4)/3=11/12<BR>\n" );
document.write( "b=11/12<BR>\n" );
document.write( "..<BR>\n" );
document.write( "We now have the information needed to write the equation of the given hyperbola.\r<BR>\n" );
document.write( "\n" );
document.write( "(y+2)^2/(11/4)^2-(x+5)^2/(11/12)^2=1\r<BR>\n" );
document.write( "\n" );
document.write( "see graph below as a visual check on the answer\r<BR>\n" );
document.write( "\n" );
document.write( "..<BR>\n" );
document.write( "y=±(7.5625+7.5625(x+5)^2/.8403)^.5-2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+300%2C+-10%2C+10%2C+-10%2C+10%2C%287.5625%2B7.5625%28x%2B5%29%5E2%2F.8403%29%5E.5-2%2C-%287.5625%2B7.5625%28x%2B5%29%5E2%2F.8403%29%5E.5-2%2C-3x-17%2C3x%2B13%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );