document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186028>Question 1186028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the ellipse whose center is at (2, 7), having a focus at (2, 10), and having an endpoint of its minor axis at (4, 7). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849482 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=849482\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the equation of the ellipse:\r<BR>\n" );
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document.write( "**1. Identify the center:**\r<BR>\n" );
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document.write( "The center of the ellipse is given as (h, k) = (2, 7).\r<BR>\n" );
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document.write( "**2. Determine the orientation:**\r<BR>\n" );
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document.write( "Since the focus (2, 10) and the center (2, 7) have the same x-coordinate, the major axis is vertical.\r<BR>\n" );
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document.write( "**3. Find the distance from the center to a focus (c):**\r<BR>\n" );
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document.write( "c = |10 - 7| = 3\r<BR>\n" );
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document.write( "**4. Find the length of the semi-minor axis (b):**\r<BR>\n" );
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document.write( "The endpoint of the minor axis is (4, 7). The distance from the center (2, 7) to this endpoint is the length of the semi-minor axis (b).\r<BR>\n" );
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document.write( "b = |4 - 2| = 2\r<BR>\n" );
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document.write( "**5. Find the length of the semi-major axis (a):**\r<BR>\n" );
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document.write( "We know that a² = b² + c².  Therefore:\r<BR>\n" );
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document.write( "a² = 2² + 3²<BR>\n" );
document.write( "a² = 4 + 9<BR>\n" );
document.write( "a² = 13<BR>\n" );
document.write( "a = √13\r<BR>\n" );
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document.write( "**6. Write the equation of the ellipse:**\r<BR>\n" );
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document.write( "The standard form equation of an ellipse with a vertical major axis is:\r<BR>\n" );
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document.write( "((x - h)² / b²) + ((y - k)² / a²) = 1\r<BR>\n" );
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document.write( "Substituting the values we found:\r<BR>\n" );
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document.write( "((x - 2)² / 2²) + ((y - 7)² / (√13)²) = 1\r<BR>\n" );
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document.write( "Simplifying:\r<BR>\n" );
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document.write( "((x - 2)² / 4) + ((y - 7)² / 13) = 1\r<BR>\n" );
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document.write( "Therefore, the equation of the ellipse is $\boxed{\frac{(x-2)^2}{4} + \frac{(y-7)^2}{13} = 1}$.<BR>\n" );
document.write( " </SPAN>\n" );
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