document.write( "Question 532249: 16y^2-9x^2+18x+64y-89=0 \n" ); document.write( "
Algebra.Com's Answer #350807 by Edwin McCravy(20060)\"\" \"About 
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document.write( "16y² - 9x² + 18x + 64y - 89 = 0\r\n" );
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document.write( "That's a hyberbola because it has both x² and y² terms with opposite\r\n" );
document.write( "signs when on the same side of the equation:\r\n" );
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document.write( "We have to get it to looking like this which opens upward and downward:\r\n" );
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document.write( "      \"%28x+-+h%29%5E2%2Fa%5E2\" - \"%28y+-+k%29%5E2%2Fb%5E2\" = 1 \r\n" );
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document.write( "or this, which opens downward or upward:\r\n" );
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document.write( "      \"%28y+-+k%29%5E2%2Fa%5E2\" - \"%28x+-+h%29%5E2%2Fb%5E2\" = 1\r\n" );
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document.write( "        16y² - 9x² + 18x + 64y - 89 = 0\r\n" );
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document.write( "Get they y terms and the x terms together and the \r\n" );
document.write( "constant on the right:\r\n" );
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document.write( "             16y² + 64y - 9x² + 18x = 89\r\n" );
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document.write( "Factor the coefficient of y², which is 16, out of the first two terms.\r\n" );
document.write( "Factor the coefficient of x², which is -9, out of the next two terms.\r\n" );
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document.write( "           16(y² + 4y) - 9(x² - 2x) = 89\r\n" );
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document.write( "Multiply the coefficient of y, which is 4, by \"1%2F2\", getting 2.\r\n" );
document.write( "Then square 2, getting +4, add +4 inside the first parentheses which\r\n" );
document.write( "is multiplied by 16 which amounts to multiplying 16·4, so we add\r\n" );
document.write( "16·4 to the right side:\r\n" );
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document.write( "Multiply the coefficient of x, which is -2, by \"1%2F2\", getting -1.\r\n" );
document.write( "Then square -1, getting +1, add +1 inside the second parentheses which\r\n" );
document.write( "is multiplied by -9 which amounts to multiplying -9·1, so we add\r\n" );
document.write( "-9·1 to the right side:\r\n" );
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document.write( "   16(y² + 4y + 4) - 9(x² - 2x + 1) = 89 + 16·4 - 9·1\r\n" );
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document.write( "Factor the trinomials in the parentheses and do some work on the right\r\n" );
document.write( "side.\r\n" );
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document.write( " 16(y + 2)(y + 2) - 9(x - 1)(x - 1) = 89 + 64 - 9\r\n" );
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document.write( "Write the factorizations as the square of binomials and finish the \r\n" );
document.write( "right side:\r\n" );
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document.write( "             16(y + 2)² - 9(x - 1)² = 144\r\n" );
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document.write( "                    \"16%28y+%2B+2%29%5E2%2F144\" - \"9%28x+-+1%29%5E2%2F144\" = \"144%2F144\"\r\n" );
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document.write( "                       \"%28y+%2B+2%29%5E2%2F9\" - \"%28x+-+1%29%5E2%2F16\" = 1 \r\n" );
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document.write( "This compares to:\r\n" );
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document.write( "                       \"%28y+-+k%29%5E2%2Fa%5E2\" - \"%28x+-+h%29%5E2%2Fb%5E2\" = 1\r\n" );
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document.write( "So it opens upward and downward.\r\n" );
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document.write( "            h = 1,  k = -2, a² = 9, so a = 3.  b² = 16, so b = 4.\r\n" );
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document.write( "The center is (h,k) = (1,-2).   \r\n" );
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document.write( "a = 3, so the transverse axis is 2·a or 6\r\n" );
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document.write( "b = 4, so the conjugate axis is 2·b or 8.\r\n" );
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document.write( "We draw the transverse axis vertically and the conjugate axis\r\n" );
document.write( "horizontally, perpendicularly bisecting each other at the center (1,-2)\r\n" );
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document.write( "Draw the defining rectangle around that cross, which is the rectangle\r\n" );
document.write( "with horizontal and vertical sides with the ends of the transverse and\r\n" );
document.write( "vertical axes bisecting the sides:\r\n" );
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document.write( "Draw and extend the diagonals of that rectangle:\r\n" );
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document.write( "Sketch in the hyperbola:\r\n" );
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document.write( "The equations of the asymptotes can be found use point-slope form:\r\n" );
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document.write( "y = \"3%2F4\"x - \"11%2F4\"\r\n" );
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document.write( "and\r\n" );
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document.write( "y = \"-3%2F4\"x - \"5%2F4\"\r\n" );
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document.write( "The foci (or focal points) are c units on each side of the center\r\n" );
document.write( "inside the two branches of the hyperbola, where\r\n" );
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document.write( "c² = a² + b²\r\n" );
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document.write( "c² = 3² + 4²\r\n" );
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document.write( "c² = 9 + 16\r\n" );
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document.write( "c² = 25\r\n" );
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document.write( "c = 5 \r\n" );
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document.write( "So they are 5 units above and below the center and are the\r\n" );
document.write( "two points inside the two branches (1,-7) and (1,3).\r\n" );
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document.write( "Edwin
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