document.write( "Question 1199474: find the following:Semi-Transverse length a, Semi-Conjugate length b,Vertices,Co-Vertices and Foci of 9x^2-4y^2-36x+8y=4 \n" ); document.write( "
Algebra.Com's Answer #833381 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I'll do this one instead.  It's done exactly the same as yours step by step:\r\n" );
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document.write( "\"25x%5E2-16y%5E2-200x%2B96y=144\"\r\n" );
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document.write( "Rearrange to get the terms in x together and the terms in y together.\r\n" );
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document.write( "\"25x%5E2-200x-16y%5E2%2B96y=144\"\r\n" );
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document.write( "Factor 25 out of the 1st and 2nd terms on the left, and factor -16 out of the\r\n" );
document.write( "3rd and 4th terms on the left:\r\n" );
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document.write( "\"25%28x%5E2-8x%29-16%28y%5E2%2B6y%29=144\"\r\n" );
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document.write( "Skip a big space at the end of each parentheses because we're going to insert\r\n" );
document.write( "a couple of terms in each one to complete the square.\r\n" );
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document.write( "\"25%28x%5E2-8x%2B%22_______%22%29-16%28y%5E2%2B6y%2B%22_______%22%29=144\"\r\n" );
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document.write( "In the first parentheses:\r\n" );
document.write( "Multiply the coefficient of x, which is -8, by 1/2 getting -4.\r\n" );
document.write( "Square -4, getting (-4)2 or +16, so we add and subtract 16 in the\r\n" );
document.write( "first blank, that is, put +16-16 in the first blank:\r\n" );
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document.write( "\"25%28x%5E2-8x%2B16-16%29-16%28y%5E2%2B6y%2B%22_______%22%29=144\"\r\n" );
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document.write( "In the second parentheses:\r\n" );
document.write( "Multiply the coefficient of y, which is -6, by 1/2 getting -3.\r\n" );
document.write( "Square -3, getting (-3)2 or +9, so we add and subtract 9 in the\r\n" );
document.write( "second blank, that is, put +9-9 in the second blank:\r\n" );
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document.write( "\"25%28x%5E2-8x%2B16-16%29-16%28y%5E2%2B6y%2B9-9%29=144\"\r\n" );
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document.write( "Factor the first three terms in each parentheses:\r\n" );
document.write( "            \"x%5E2-8x%2B16=%28x-4%29%28x-4%29=%28x-4%29%5E2\"\r\n" );
document.write( "            \"y%5E2-6y%2B9=%28x-3%29%28x-3%29=%28x-3%29%5E2\"    \r\n" );
document.write( " \r\n" );
document.write( "\"25%28%28x-4%29%5E2-16%29-16%28%28x-3%29%5E2-9%29=144\"\r\n" );
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document.write( "Distribute the 25 and the -16, being careful to leave the squared parentheses intact:\r\n" );
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document.write( "\"25%28x-4%29%5E2-400-16%28x-3%29%5E2%2B144=144\"\r\n" );
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document.write( "\"25%28x-4%29%5E2-16%28x-3%29%5E2-256=144\"\r\n" );
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document.write( "\"25%28x-4%29%5E2-16%28x-3%29%5E2=400\"\r\n" );
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document.write( "Divide every term by 400 to get 1 on the right side:\r\n" );
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document.write( "\"25%28x-4%29%5E2%2F400-16%28x-3%29%5E2%2F400=400%2F400\"\r\n" );
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document.write( "cancel:\r\n" );
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document.write( "\"%28x-4%29%5E2%2F16-%28x-3%29%5E2%2F25=1\"\r\n" );
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document.write( "Compare to\r\n" );
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document.write( "\"%28x-h%29%5E2%2Fa%5E2-%28x-k%29%5E2%2Fb%5E2=1\"\r\n" );
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document.write( "Where the center is (h,k) = (4,3),\r\n" );
document.write( "semi-transverse axis length = a = 4, \r\n" );
document.write( "semi-conjugate axis length = b = 5\r\n" );
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document.write( "Plot the center (h,k) = (4,3).\r\n" );
document.write( "Draw the transverse axis horizontally from the center a=4 units \r\n" );
document.write( "right and a=4 units left.\r\n" );
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document.write( "Draw the conjugate axis vertically from the center b=5 units \r\n" );
document.write( "upward and b=5 units downward.\r\n" );
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document.write( "The endpoints of the transverse axes are the two vertices.\r\n" );
document.write( "From the graph, we see they are (0,3), and (8,3).\r\n" );
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document.write( "The endpoints of the conjugate axes are the two co-vertices.\r\n" );
document.write( "From the graph, we see they are (4,-2), and (4,8).\r\n" );
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document.write( "Next, we draw the defining rectangle around the transverse\r\n" );
document.write( "and conjugate axes like this.  And we draw and extend the\r\n" );
document.write( "two diagonals of the defining rectangle:\r\n" );
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document.write( "Now we can sketch in the hyperbola approching the two ansymptotes:\r\n" );
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document.write( "Edwin

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