document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1109030>Question 1109030</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 16x²+ 9y²+64x-54y+1=0 center </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #724066 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=724066\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> looks like the equation of an ellipse.\r<BR>\n" );
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document.write( "assuming that's what it is, then transform the equation into standard form of an ellipse.\r<BR>\n" );
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document.write( "start with 16x^2 + 9y^2 + 64x - 54y + 1 = 0\r<BR>\n" );
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document.write( "subtract 1 from both sides to get 16x^2 + 9y^2 + 64x - 54y = -1\r<BR>\n" );
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document.write( "group the x's and the y's together to get (16x^2 + 64x) + (9y^2 - 54y) = -1\r<BR>\n" );
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document.write( "factor out the coefficients so that the grouped variables start with a coefficient of 1.\r<BR>\n" );
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document.write( "you will get 16(x^2 + 4x) + 9(y^2 - 6y) = -1\r<BR>\n" );
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document.write( "complete the squares on (x^2 + 4x) to get (x+2)^2 - 4<BR>\n" );
document.write( "complete the squares on (y^2 - 6y) to get (y-3)^2 - 9\r<BR>\n" );
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document.write( "equation becomes 16 * ((x+2)^2 - 4) + 9 * ((y-3)^2 - 9) = -1\r<BR>\n" );
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document.write( "simplify to get 16 * (x+2)^2 - 64 + 9 * (y-3)^2 - 81 = -1\r<BR>\n" );
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document.write( "add 64 and 81 to both sides of the equation to get:\r<BR>\n" );
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document.write( "16 * (x+2)^2 + 9 * (y-3)^2 = 144\r<BR>\n" );
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document.write( "divide both sides of the equation by 144 to get:\r<BR>\n" );
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document.write( "16 * (x+2)^2 / 144 + 9 * (y-3)^2 / 144 = 1\r<BR>\n" );
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document.write( "this can be written as 16/144 * (x+2)^2 + 9/144 * (y-3)^2 = 1\r<BR>\n" );
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document.write( "simplify so that the numerator in the fractions is 1 to get (x+2)^2 / 9 + (y-3)^2 / 16 = 1\r<BR>\n" );
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document.write( "this is now in standard form of (x-h)^2 / b^2 + (y-k)^2 / a^2 = 1\r<BR>\n" );
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document.write( "the designation of the letter a always goes where the largest denominator is and the designation of the letter b always goes where the smallest denominator is.\r<BR>\n" );
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document.write( "that's why the b^2 went under the (x-h)^2 and the a^2 went under the (y-k)^2.\r<BR>\n" );
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document.write( "in your equation, you have:\r<BR>\n" );
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document.write( "a^2 = 16 which makes a = 4<BR>\n" );
document.write( "b^2 = 9 which makes b = 3<BR>\n" );
document.write( "center of the ellipse is (h,k) which make the center (-2,3).\r<BR>\n" );
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document.write( "the major axis of the ellipse is the axis that is the longest.\r<BR>\n" );
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document.write( "a is the distance along the major axis from the center of the ellipse to the vertex of the ellipse.\r<BR>\n" );
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document.write( "b is the distance along the minor axis from the center of the ellipse to the co-vertex of the ellipse.\r<BR>\n" );
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document.write( "c is the distance along the major axis from the center of the ellipse to the the focus of the ellipse.\r<BR>\n" );
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document.write( "c is calculated using the formula c^2 = a^2 - b^2.\r<BR>\n" );
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document.write( "in your equation, c^2 is therefore equal to 9 - 4 = 5.\r<BR>\n" );
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document.write( "this makes c = sqrt(5).\r<BR>\n" );
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document.write( "your important values to the ellipse are now:\r<BR>\n" );
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document.write( "a = 4<BR>\n" );
document.write( "b = 3<BR>\n" );
document.write( "c = sqrt(5)<BR>\n" );
document.write( "center = (-2,3)\r<BR>\n" );
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document.write( "the graph of your ellipse looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/021002.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "here's a reference that should help you understand what's going on.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/ellipse.htm\" target = \"_blank\">http://www.purplemath.com/modules/ellipse.htm</a>\r<BR>\n" );
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document.write( "here's my diagram of the main parts of the ellipse that should help you identify what's what on the ellipse.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/021003.jpg\" alt=\"$$$\" >\r<BR>\n" );
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