document.write( "Question 957500: Hi,
\n" ); document.write( "I'm supposed to write the equation of an ellipse with foci at (4,1) and (4,-7) and vertices at (4,2) and (4,-8) but I can't figure out what to do after the plus sign. I know the top is (x-4)^2 but I don't know how to figure out the bottom.
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Algebra.Com's Answer #585115 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "To do ellipse problems, you must ALWAYS draw the graph.\r\n" );
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document.write( "You must also know all the parts of an ellipse, and the\r\n" );
document.write( "equations and formulas.\r\n" );
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document.write( "They are:\r\n" );
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document.write( "1. The center, which has coordinates (h,k).\r\n" );
document.write( "2. The major axis = the longesr chord that passes through the center,\r\n" );
document.write( "bisected at the center. Its endpoints are the two vertices.\r\n" );
document.write( "3. The two vertices \"a\" units from the center.\r\n" );
document.write( "4. The two covertices \"b\" units from the center.\r\n" );
document.write( "5. The semimajor axes = \"a\" units long, from the center to each vertex. \r\n" );
document.write( "6. The minor axis = the shortest chord that passes through the center,\r\n" );
document.write( "bisected at the center. Its endpoints are the two covertices.\r\n" );
document.write( "7. The semiminor axes = \"b\" units long, from the center to each vertex \r\n" );
document.write( "8. The two foci = points on the major axis \"c\" units from the center.\r\n" );
document.write( "9. The formula for \"c\", which is c2 = a2 - b2  \r\n" );
document.write( "10. The standard equation for an ellipse that looks like this: \"drawing%2820%2C10%2C-2%2C2%2C-1%2C1%2Carc%280%2C0%2C-3.9%2C1.9%29+%29\" which is \"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2\"\"%22%22=%22%22\"\"1\"\r\n" );
document.write( "11. The standard equation for an ellipse that looks like this: \"drawing%2810%2C20%2C-1%2C1%2C-2%2C2%2Carc%280%2C0%2C1.9%2C-3.9%29+%29\" which is \"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "We begin by plotting the given parts: the vertices and the foci:\r\n" );
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document.write( "The center is the midpoint between the two vertices, which is also\r\n" );
document.write( "the midpoint between the two foci. By looking at the graph we see that it is\r\n" );
document.write( "the point (4,-3).  So we plot the center (4,-3), in red below:\r\n" );
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document.write( "We see that a = 5 because we observe that it is 5 units from the center to\r\n" );
document.write( "each vertex.\r\n" );
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document.write( "We draw the major axis, from vertex to vertex (in green). It is 10\r\n" );
document.write( "units long, a = 5 units on each side of the center\r\n" );
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document.write( "To draw the minor axis we must calculate b, using the formula \r\n" );
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document.write( "c2 = a2 - b2\r\n" );
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document.write( "We know that the semimajor axis is 5 units long, so a = 5. \r\n" );
document.write( "We know that c = 4, because we observe that it is 4 units from the center\r\n" );
document.write( "to each focus.  We substitute these:\r\n" );
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document.write( " 42 = 52 - b2\r\n" );
document.write( " 16 = 25 - b2\r\n" );
document.write( " -9 = -b2 \r\n" );
document.write( "  9 = b2\r\n" );
document.write( "  3 = b\r\n" );
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document.write( "Now we can draw the minor axis  (also in green) perpendicular to\r\n" );
document.write( "the major axis bisected at the center.  It is 6 units long, b = 3\r\n" );
document.write( "units on each side of the center.  It is the line that connects \r\n" );
document.write( "the two covertices:\r\n" );
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document.write( "Now its easy to sketch in the ellipse:\r\n" );
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document.write( "Now since we see that it is an ellipse that looks like this:\"drawing%2810%2C20%2C-1%2C1%2C-2%2C2%2Carc%280%2C0%2C1.9%2C-3.9%29+%29\"\r\n" );
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document.write( "it has the standard equation:\r\n" );
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document.write( " \"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "And we substitute (h,k) = (4,-3), a = 5, and b = 3,\r\n" );
document.write( "and the standard equation is\r\n" );
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document.write( " \"%28x-4%29%5E2%2F3%5E2%2B%28y%5E%22%22-%28-3%29%29%5E2%2F5%5E2\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( " \"%28x-4%29%5E2%2F9%2B%28y%2B3%29%5E2%2F25\"\"%22%22=%22%22\"\"1\"\r\n" );
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document.write( "Edwin
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