document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=976438>Question 976438</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find: a. center b. vertices c. foci d. extremities of the minor axis e. equation of the directrices f. eccentricity g. graph of the ellipse\r<BR>\n" );
document.write( "\n" );
document.write( "81x^2 + 4y^2 - 324x + 24y + 324 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #598022 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=598022\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 81x^2 + 4y^2 - 324x + 24y + 324 = 0<BR>\n" );
document.write( "rewrite\r<BR>\n" );
document.write( "\n" );
document.write( "81x^2- 324 x  + 4y^2  + 24 y =-324\r<BR>\n" );
document.write( "\n" );
document.write( "factor<BR>\n" );
document.write( "81(x^2-4x)  + 4(y^2 + 6y)=-324<BR>\n" );
document.write( "complete the square, adding the product of the coefficient and the last term to the right side.\r<BR>\n" );
document.write( "\n" );
document.write( "81 (x^2-4x+4) +4(y^2 +6y +9)=-324+360=36<BR>\n" );
document.write( "Divide by 36 and rewrite\r<BR>\n" );
document.write( "\n" );
document.write( "(81/36) (x-2)^2 + (4/36) (y+3)^2=1<BR>\n" );
document.write( "{[9(x-2)/]4}^2+ {(y+3)/9}^2=1<BR>\n" );
document.write( "{(x-2)/((4/9)}^2 +{(y+3)/9}^2=1<BR>\n" );
document.write( "center is at (2,-3)<BR>\n" );
document.write( "Larger denominator gives a^2 so that is 9, and a=3<BR>\n" );
document.write( "Smaller denominator is b^2, so that is (2/3).  This is an ellipse with the major axis along the y-axis.<BR>\n" );
document.write( "So vertices are at (2,0) and (2,-6)<BR>\n" );
document.write( "The semi-minor vertices are at (1 1/3,-3) and (2 2/3, -3)<BR>\n" );
document.write( "foci: b^2+c^2=a^2, or c^2=9-(4/9)=(77/9).  c^2=sqrt(77)/3<BR>\n" );
document.write( "foci will be (2,-3+sqrt(77)/3 )  and (2, -3 - sqrt (77)/3 )<BR>\n" );
document.write( "eccentricity is [sqrt(a^2-b^2)]/a = sqrt (77)/3  / 3 = sqrt (77)/9\r<BR>\n" );
document.write( "\n" );
document.write( "y=(+/-) (a/e) for equation of directrices:  They are perpendicular to major axis. <BR>\n" );
document.write( "y=(+/-) a^2/c=9/[sqrt(77)/9]<BR>\n" );
document.write( "y= (+/-) 81/sqrt (77)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );