document.write( "Question 798039: find the general equation of ellipse with center at (1,5), major axis parallel to x-axis, length of latus rectum is 9/4, distance between foci is 2squareroot of 55 units. \n" ); document.write( "

Algebra.Com's Answer #482030 by Edwin McCravy(20060) $\"\"$ $\"About$

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Find the general equation of ellipse with center at (1,5), major axis parallel to x-axis, length of latus rectum is 9/4, distance between foci is 2squareroot of 55 units.
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document.write( "We will let P and S be the foci.\r\n" );
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document.write( "For the above we are given:\r\n" );
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document.write( "O = center = (1,5)\r\n" );
document.write( "QR = TU = latus rectum = \r\n" );
document.write( "SP = distance between foci = 2√55\r\n" );
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document.write( "So we can deduce that:\r\n" );
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document.write( "OP = c = SP = √55\r\n" );
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document.write( "P = (1+√55,5)\r\n" );
document.write( "PQ = QR =  = \r\n" );
document.write( "Q = (1+√55,5+)\r\n" );
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document.write( "The standard equation of the ellipse, since it has major axis horizontal is\r\n" );
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document.write( "To avoid so many fractions we will clear of fractions:\r\n" );
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document.write( "b²(x-h)² + a²(y-k)² = a²b²\r\n" );
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document.write( "b²(x-1)² + a²(y-5)² = a²b²\r\n" );
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document.write( "Since we know that the ellipse passes through the point Q = (1+√55,5+),\r\n" );
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document.write( "b²(1+√55-1)² + a²(5+-5)² = a²b²\r\n" );
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document.write( "b²(√55)² + a²()² = a²b²\r\n" );
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document.write( "b²(55) + a²() = a²b²\r\n" );
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document.write( "Multiplying through by 64:\r\n" );
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document.write( "3520b² + 81a² = 64a²b²\r\n" );
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document.write( "We know that all ellipses have the property\r\n" );
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document.write( "c² = a²-b²\r\n" );
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document.write( "and since c = √55, c² = 55\r\n" );
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document.write( "a²-b² = 55 \r\n" );
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document.write( "So we have the system of equations:\r\n" );
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document.write( "3520b² + 81a² = 64a²b²\r\n" );
document.write( "a²-b² = 55\r\n" );
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document.write( "Solve the second equation for a²,  a²=55-b²\r\n" );
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document.write( "Substituting that in the first:\r\n" );
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document.write( "  3520b² + 81(55+b²) = 64(55+b²)b²\r\n" );
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document.write( "3520b² + 4455 + 81b² = 3520b² + 64b⁴\r\n" );
document.write( " -64b⁴ + 81b² + 4455 = 0\r\n" );
document.write( "  64b⁴ - 81b² - 4455 = 0\r\n" );
document.write( "    (b²-9)(64b²+495) = 0 \r\n" );
document.write( "     b²-9 = 0;  64b²+495 = 0\r\n" );
document.write( "       b² = 9;  64b² = -495     \r\n" );
document.write( "        b = 3;    b = imaginary (ignore)\r\n" );
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document.write( "Substitute b=3 in a²-b² = 55\r\n" );
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document.write( "a²-b² = 55\r\n" );
document.write( "a²-3² = 55\r\n" );
document.write( " a²-9 = 55\r\n" );
document.write( "   a² = 64\r\n" );
document.write( "    a = 8\r\n" );
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document.write( "If we wanted the standard equation, it would be:\r\n" );
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document.write( "But we want the general equation,\r\n" );
document.write( "not the standard, so we substitute in:\r\n" );
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document.write( "       b²(x-h)² + a²(y-k)² = a²b²\r\n" );
document.write( "        9(x-1)² + 64(y-5)² = 64·9\r\n" );
document.write( "9(x²-2x+1) + 64(y²-10y+25) = 576\r\n" );
document.write( "9x²-18x+9 + 64y²-640y+1600 = 576\r\n" );
document.write( "    9x²+64y²-18x-640y+1609 = 576\r\n" );
document.write( "    9x²+64y²-18x-640y+1033 = 0\r\n" );
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document.write( "Edwin

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