document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169000>Question 1169000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> find the equation of the ellipse if the vertex is at (5,1) and foci at (-2,1) and (4,1). express your answer in general form.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851462 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solution:<BR>\n" );
document.write( "The given information about the ellipse is:<BR>\n" );
document.write( "Vertex: $(5, 1)$<BR>\n" );
document.write( "Foci: $(-2, 1)$ and $(4, 1)$\r<BR>\n" );
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document.write( "Since the y-coordinates of the vertex and the foci are the same, the major axis of the ellipse is horizontal.<BR>\n" );
document.write( "The center of the ellipse is the midpoint of the foci:<BR>\n" );
document.write( "Center $(h, k) = \left( \frac{-2 + 4}{2}, \frac{1 + 1}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1)$\r<BR>\n" );
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document.write( "The distance between the center and a focus is $c$. Using the focus at $(4, 1)$:<BR>\n" );
document.write( "$c = \sqrt{(4 - 1)^2 + (1 - 1)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3$\r<BR>\n" );
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document.write( "The distance between the center and a vertex is $a$. Using the vertex at $(5, 1)$:<BR>\n" );
document.write( "$a = \sqrt{(5 - 1)^2 + (1 - 1)^2} = \sqrt{4^2 + 0^2} = \sqrt{16} = 4$\r<BR>\n" );
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document.write( "We know the relationship between $a$, $b$, and $c$ for an ellipse: $c^2 = a^2 - b^2$.<BR>\n" );
document.write( "Substituting the values of $a$ and $c$:<BR>\n" );
document.write( "$3^2 = 4^2 - b^2$<BR>\n" );
document.write( "$9 = 16 - b^2$<BR>\n" );
document.write( "$b^2 = 16 - 9 = 7$\r<BR>\n" );
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document.write( "The standard form of the equation of a horizontal ellipse with center $(h, k)$ is:<BR>\n" );
document.write( "$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$\r<BR>\n" );
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document.write( "Substituting the values of $h=1$, $k=1$, $a^2 = 16$, and $b^2 = 7$:<BR>\n" );
document.write( "$\frac{(x - 1)^2}{16} + \frac{(y - 1)^2}{7} = 1$\r<BR>\n" );
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document.write( "To express the equation in general form, we eliminate the denominators and expand:<BR>\n" );
document.write( "$7(x - 1)^2 + 16(y - 1)^2 = 16 \times 7$<BR>\n" );
document.write( "$7(x^2 - 2x + 1) + 16(y^2 - 2y + 1) = 112$<BR>\n" );
document.write( "$7x^2 - 14x + 7 + 16y^2 - 32y + 16 = 112$<BR>\n" );
document.write( "$7x^2 + 16y^2 - 14x - 32y + 23 = 112$<BR>\n" );
document.write( "$7x^2 + 16y^2 - 14x - 32y + 23 - 112 = 0$<BR>\n" );
document.write( "$7x^2 + 16y^2 - 14x - 32y - 89 = 0$\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{7x^2 + 16y^2 - 14x - 32y - 89 = 0}$ </SPAN>\n" );
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