document.write( "Question 1073479: How do I solve and graph the equation 25x^2-16y^2-100x+96y+276=0 \n" ); document.write( "
Algebra.Com's Answer #688308 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "25x^2-16y^2-100x+96y+276=\r\n" );
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document.write( "The object is to make it look like this:\r\n" );
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document.write( " (x-h)²  (y-k)²    \r\n" );
document.write( "—————— - ——————— = 1\r\n" );
document.write( "   a²      b²\r\n" );
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document.write( "which is a hyperbola that looks like this )( or:\r\n" );
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document.write( " (y-k)²  (x-h)²    \r\n" );
document.write( "—————— + ——————— = 1\r\n" );
document.write( "   a²      b²\r\n" );
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document.write( "Which has one branch opening upward and the other downward\r\n" );
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document.write( "We start with this:\r\n" );
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document.write( "       25x² - 16y² - 100x + 96y + 276 = 0\r\n" );
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document.write( "We get the 276 on the other side as a -276:\r\n" );
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document.write( "             25x² - 16y² - 100x + 96y = -276\r\n" );
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document.write( "We have to switch the two middle terms on the\r\n" );
document.write( "left so that the terms are in the order \"x², x, y², y\".\r\n" );
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document.write( "             25x² - 100x - 16y² + 96y = -276\r\n" );
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document.write( "Write it this way: \r\n" );
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document.write( "        [25x² - 100x] - [16y² - 96y] = -276\r\n" );
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document.write( "The coefficient of x² is 25, so let's factor that out\r\n" );
document.write( "in the 1st bracket.  \r\n" );
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document.write( "        [25(x² - 4x)] - [16y² - 96y] = -276\r\n" );
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document.write( "The coefficient of y² is 16, so let's factor that out\r\n" );
document.write( "in the 2nd bracket.  \r\n" );
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document.write( "       [25(x² - 4x)] - [16(y² - 6y)] = -276\r\n" );
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document.write( "Now we'll dispense with the brackets and just have parentheses:\r\n" );
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document.write( "           25(x² - 4x) - 16(y² - 6y) = -276\r\n" );
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document.write( "Next we want to make those two binomials into trinomials.\r\n" );
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document.write( "We skip some space after those binomials \r\n" );
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document.write( "   25(x² - 4x    ) - 16(y² - 6y    ) = -276\r\n" );
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document.write( "so we can add a number in those two spaces to make those \r\n" );
document.write( "binomials into trinomials so they'll factor into squares \r\n" );
document.write( "of binomials.\r\n" );
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document.write( "Now let's figure out what number goes in the first space.\r\n" );
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document.write( "The coefficient of x is -4 so we take half of it, getting -2,\r\n" );
document.write( "then we square -2, getting (-2)² or 4, but wait!  See the 25 in \r\n" );
document.write( "front of the first parentheses?  If we put a 4 in that space,\r\n" );
document.write( "It will get multiplied by the 25 in front of the parentheses.\r\n" );
document.write( "In other words putting a 4 in that first space will in effect \r\n" );
document.write( "amount to the same as adding 25 times 4 or 100 to the left side,\r\n" );
document.write( "not just 4.  So we have to add 25(4) to the right side to offset\r\n" );
document.write( "adding 4 inside that parentheses on the left since it will be \r\n" );
document.write( "multiplied by the 25, so we add 4 in the first space, but \r\n" );
document.write( "we have to add 100 to the other side of the equation:\r\n" );
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document.write( "   25(x² - 4x + 4) - 16(y² - 6y    ) = -276 + 100\r\n" );
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document.write( "Now let's figure out what number goes in the second space.\r\n" );
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document.write( "The coefficient of y is -6 so we take half of it, getting -3,\r\n" );
document.write( "then we square -3, getting (-3)² or 9, but wait!  See the -16 in \r\n" );
document.write( "front of the second parentheses?  If we put a 9 in that space,\r\n" );
document.write( "It will get multiplied by the -16 in front of the parentheses.\r\n" );
document.write( "In other words putting a 9 in that second box will in effect \r\n" );
document.write( "amount to the same as adding -16 times 9 or -144 to the left side,\r\n" );
document.write( "not just 9.  So we have to add -144 to the right side to offset\r\n" );
document.write( "adding 9 inside that parentheses since it will be multiplied\r\n" );
document.write( "by the -16, so we have:\r\n" );
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document.write( "   25(x² - 4x + 4) - 16(y² - 6y + 9) = -276 + 100 - 144\r\n" );
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document.write( "Notice that what's in the first parentheses,\r\n" );
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document.write( "x²-4x+4 factors as (x-2)(x-2) or (x-2)²\r\n" );
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document.write( "Also notice that what's in the second parentheses\r\n" );
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document.write( "y²-6y+9 factors as (y-3)(y-3) or (y-3)².\r\n" );
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document.write( "So this\r\n" );
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document.write( "   25(x² - 4x + 4) - 16(y² - 6y + 9) = -276 + 100 - 144\r\n" );
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document.write( "becomes this\r\n" );
document.write( "   \r\n" );
document.write( "                   25(x-2)² - 16(y-3)² = -320\r\n" );
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document.write( "after substituting their factorization for the parentheses\r\n" );
document.write( "and combining the terms on the right.\r\n" );
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document.write( "Next we get a 1 on the right by dividing all three terms by -320:\r\n" );
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document.write( " 25(x-2)²   16(y-3)²    -320\r\n" );
document.write( "————————— - ———————— = —————— \r\n" );
document.write( "  -320       -320       -320\r\n" );
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document.write( "Divide top and bottom of the first term by 25\r\n" );
document.write( "Divide top and bottom of the first term by 16\r\n" );
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document.write( " (x-2)²     (y-3)²    \r\n" );
document.write( "———————— - ———————— = 1\r\n" );
document.write( "-320/25      -20       \r\n" );
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document.write( "Simplifying the signs:\r\n" );
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document.write( "   (x-2)²     (y-3)²    \r\n" );
document.write( "- ———————— + ———————— = 1\r\n" );
document.write( "   320/25      20  \r\n" );
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document.write( "Reducing the fraction on the bottom of the first term:\r\n" );
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document.write( "   (x-2)²     (y-3)²    \r\n" );
document.write( "- ———————— + ———————— = 1\r\n" );
document.write( "    64/5       20\r\n" );
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document.write( "Let's write the positive term first:\r\n" );
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document.write( "    (y-3)²   (x-2)²    \r\n" );
document.write( "   —————— - ——————— = 1\r\n" );
document.write( "     20      64/5\r\n" );
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document.write( "which is in the form:\r\n" );
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document.write( "    (y-k)²  (x-h)²    \r\n" );
document.write( "   —————— + ——————— = 1\r\n" );
document.write( "      a²      b²\r\n" );
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document.write( "So the hyperbola has one branch opening upward\r\n" );
document.write( "and the other downward.\r\n" );
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document.write( "We now have h=2, k=3, a²=20, b²=64/5.\r\n" );
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document.write( "The center is (h,k) = (2,3)\r\n" );
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document.write( "Plot it:\r\n" );
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document.write( "                    __     _ \r\n" );
document.write( "Since a² = 20, a = √20 = 2√5\r\n" );
document.write( "                        _     _\r\n" );
document.write( "Since b² = 64/5, b = 8/√5 = 8√5/5\r\n" );
document.write( "      _\r\n" );
document.write( "a = 2√5 is the semi-transverse axis's length, so draw the vertical\r\n" );
document.write( "transverse axis 2a or 4√5 units long with the center as the midpoint.\r\n" );
document.write( "We also draw the horizontal conjugate axis 2b or 16√5/5 units long \r\n" );
document.write( "with the center as the midpoint:\r\n" );
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document.write( "Now we draw the defining rectangle with the ends of the transverse\r\n" );
document.write( "and conjugate axes as midpoints of the sides:\r\n" );
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document.write( "Now draw and extend the diagonals of the defining rectangle\r\n" );
document.write( "which are the asymptotes of the hyperbola:\r\n" );
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document.write( "Now we can sketch in the hyperbola:\r\n" );
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document.write( "Edwin
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