document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=475095>Question 475095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I put this equation in standard hyperbola form AND then find the coordinates of the foci? My biggest problem is putting it in standard hyperbola form because I am bad at factoring!\r<BR>\n" );
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document.write( "-16x^2 + 25y^2 - 32x - 250y + 209 = 0 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #325806 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20059)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=325806\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "-16x² + 25y² - 32x - 250y + 209 = 0\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( "      -16x² + 25y² - 32x - 250y + 209 = 0\r\n" );
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document.write( "We get the 209 on the other side as a -209:\r\n" );
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document.write( "            -16x² + 25y² - 32x - 250y = -209\r\n" );
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document.write( "Ee 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( "            -16x² - 32x + 25y² - 250y = -209\r\n" );
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document.write( "Write it this way: \r\n" );
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document.write( "        [-16x² - 32x] + [25y² - 250y] = -209\r\n" );
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document.write( "The coefficient of x² is -16, so let's factor that out\r\n" );
document.write( "in the 1st bracket.  (Remember to change the sign when\r\n" );
document.write( "factoring out a negative.  That's why we have +2x \r\n" );
document.write( "and not -2x]:\r\n" );
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document.write( "        [-16(x² + 2x)] + [25y² - 250y] = -209\r\n" );
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document.write( "The coefficient of y² is 15, so let's factor that out\r\n" );
document.write( "in the 2nd bracket.  \r\n" );
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document.write( "       [-16(x² + 2x)] + [25(y² - 10y)] = -209\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( "           -16(x² + 2x) + 25(y² - 10y) = -209\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( "   -16(x² + 2x    ) + 25(y² - 10y    ) = -209\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 2 so we take half of it, getting 1,\r\n" );
document.write( "then we square 1, getting 1² or 1, but wait!  See the -16 in \r\n" );
document.write( "front of the first parentheses?  If we put a 1 in that box,\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 1 in that first space will in effect \r\n" );
document.write( "amount to the same as adding -16 times 1 or -16 to the left side,\r\n" );
document.write( "not just 1.  So we have to add -16(1) to the right side to offset\r\n" );
document.write( "adding 1 inside that parentheses on the left since it will be \r\n" );
document.write( "multiplied by the -16, so we so we add 1 in the first space, but \r\n" );
document.write( "we have to add -16 to the other side of the equation:\r\n" );
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document.write( "  -16(x² + 2x + 1) + [25(y² - 10y    ) = -209 - 16(1)\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 -10 so we take half of it, getting -5,\r\n" );
document.write( "then we square -5, getting (-5)² or 25, but wait!  See the 25 in \r\n" );
document.write( "front of the second parentheses?  If we put a 25 in that box,\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 25 in that second box will in effect \r\n" );
document.write( "amount to the same as adding 25 times 25 or 625 to the left side,\r\n" );
document.write( "not just 25.  So we have to add 25(25) to the right side to offset\r\n" );
document.write( "adding 25 inside that parentheses since it will be multiplied\r\n" );
document.write( "by the 25, so we have:\r\n" );
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document.write( "  -16(x² + 2x + 1) + [25(y² - 10y + 25) = -209 - 16(1) + 25(25)\r\n" );
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document.write( "Notice that what's in the first parentheses,\r\n" );
document.write( "x²+2x+1 factors as (x-1)(x-1) or (x-1)²\r\n" );
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document.write( "Also notice that what's in the second parentheses\r\n" );
document.write( "y²-10y+25 factors as (y-5)(y-5) or (y+1)².\r\n" );
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document.write( "So this\r\n" );
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document.write( "   -16(x² + 2x + 1) + 25(y² - 10y + 25) = -209 - 16(1) + 25(25)\r\n" );
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document.write( "becomes this\r\n" );
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document.write( "                   -16(x+1)² + 25(y-5)² = 400\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 400:\r\n" );
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document.write( "-16(x+1)²   25(y-5)²    400\r\n" );
document.write( "————————— + ———————— = ————— \r\n" );
document.write( "   400        400       400\r\n" );
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document.write( "And that simplifies to:\r\n" );
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document.write( "    (x+1)²   (y-5)²    \r\n" );
document.write( " - —————— + ——————— = 1\r\n" );
document.write( "     25       16       \r\n" );
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document.write( "Let's write the positive term first:\r\n" );
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document.write( "    (y-5)²   (x+1)²    \r\n" );
document.write( "   —————— - ——————— = 1\r\n" );
document.write( "     16       25\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=-1, k=5, a²=16, b²=25.\r\n" );
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document.write( "The center is (h,k) = (-1,5)\r\n" );
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document.write( "Plot it:\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-11%2C9%2C-5%2C15%2C%0D%0A%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%29%2C+circle%28-1%2C5%2C.1%29%2C%0D%0Alocate%28-3.5%2C5%2C%22%28-1%2C5%29%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "Since a² = 16, a = 4\r\n" );
document.write( "Since b² = 25, b = 5\r\n" );
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document.write( "a = 4 is the semi-transverse axis's length, so draw the vertical\r\n" );
document.write( "transverse axis 2a or 8 units long with the center as the midpoint.\r\n" );
document.write( "We also draw the horizontal conjugate axis 2b or 10 units long \r\n" );
document.write( "with the center as the midpoint:\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-11%2C9%2C-5%2C15%2C%0D%0Agreen%28line%28-6%2C5%2C4%2C5%29%2Cline%28-1%2C1%2C-1%2C9%29%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%29%2C+circle%28-1%2C5%2C.1%29%2C%0D%0Alocate%28-3.5%2C5%2C%22%28-1%2C5%29%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-11%2C9%2C-5%2C15%2C%0D%0Agreen%28line%28-6%2C5%2C4%2C5%29%2Cline%28-1%2C1%2C-1%2C9%29%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%29%2C+circle%28-1%2C5%2C.1%29%2C%0D%0Aline%28-6%2C1%2C-6%2C9%29%2Cline%284%2C1%2C4%2C9%29%2Cline%28-6%2C1%2C4%2C1%29%2Cline%28-6%2C9%2C4%2C9%29+%2C%0D%0Alocate%28-3.5%2C5%2C%22%28-1%2C5%29%22%29+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\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( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-11%2C9%2C-5%2C15%2C%0D%0Agreen%28line%28-6%2C5%2C4%2C5%29%2Cline%28-1%2C1%2C-1%2C9%29%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%29%2C+circle%28-1%2C5%2C.1%29%2C%0D%0Aline%28-6%2C1%2C-6%2C9%29%2Cline%284%2C1%2C4%2C9%29%2Cline%28-6%2C1%2C4%2C1%29%2Cline%28-6%2C9%2C4%2C9%29%2C%0D%0Alocate%28-3.5%2C5%2C%22%28-1%2C5%29%22%29%2C+green%28line%28-16%2C-7%2C14%2C17%29%2Cline%28-16%2C17%2C19%2C-11%29%29+++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "Now we can sketch in the hyperbola:\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-11%2C9%2C-5%2C15%2C%0D%0Agreen%28line%28-6%2C5%2C4%2C5%29%2Cline%28-1%2C1%2C-1%2C9%29%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%2C5%2B4sqrt%281%2B%28x%2B1%29%5E2%2F25%29%29%2C+circle%28-1%2C5%2C.1%29%2C%0D%0Aline%28-6%2C1%2C-6%2C9%29%2Cline%284%2C1%2C4%2C9%29%2Cline%28-6%2C1%2C4%2C1%29%2Cline%28-6%2C9%2C4%2C9%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%2C5-4sqrt%281%2B%28x%2B1%29%5E2%2F25%29%29%2C%0D%0Alocate%28-3.5%2C5%2C%22%28-1%2C5%29%22%29%2C+green%28line%28-16%2C-7%2C14%2C17%29%2Cline%28-16%2C17%2C19%2C-11%29%29+++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "The vertices are (-1,1) and (-1,9)\r\n" );
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document.write( "We only need to find the coordinates of the two foci.\r\n" );
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document.write( "The foci are points which are c units from the center and colinear\r\n" );
document.write( "with the vertices and the center:\r\n" );
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document.write( "We calculate c from:\r\n" );
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document.write( "c² = a² + b²\r\n" );
document.write( "c² = 4² + 5²\r\n" );
document.write( "c² = 16 + 25\r\n" );
document.write( "c² = 41\r\n" );
document.write( " c = &#8730;41 about 6.4\r\n" );
document.write( "                        __              __\r\n" );
document.write( "So the foci are F(-1,5+&#8730;41) and F(-1,5-&#8730;41).\r\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28400%2C400%2C-11%2C9%2C-5%2C15%2C%0D%0Agreen%28line%28-6%2C5%2C4%2C5%29%2Cline%28-1%2C1%2C-1%2C9%29%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%2C5%2B4sqrt%281%2B%28x%2B1%29%5E2%2F25%29%29%2C+circle%28-1%2C5%2C.1%29%2C%0D%0Aline%28-6%2C1%2C-6%2C9%29%2Cline%284%2C1%2C4%2C9%29%2Cline%28-6%2C1%2C4%2C1%29%2Cline%28-6%2C9%2C4%2C9%29%2C%0D%0Agraph%28400%2C400%2C-11%2C9%2C-5%2C15%2C5-4sqrt%281%2B%28x%2B1%29%5E2%2F25%29%29%2C%0D%0Alocate%28-3.5%2C5%2C%22%28-1%2C5%29%22%29%2C+green%28line%28-16%2C-7%2C14%2C17%29%2Cline%28-16%2C17%2C19%2C-11%29%29%2C%0D%0Acircle%28-1%2C5%2Bsqrt%2841%29%2C.2%29%2Ccircle%28-1%2C5%2Bsqrt%2841%29%2C.15%29%2Ccircle%28-1%2C5%2Bsqrt%2841%29%2C.1%29%2C%0D%0Acircle%28-1%2C5-sqrt%2841%29%2C.2%29%2Ccircle%28-1%2C5-sqrt%2841%29%2C.15%29%2Ccircle%28-1%2C5-sqrt%2841%29%2C.1%29%2C%0D%0Alocate%28-6.5%2C5%2Bsqrt%2841%29%2B.5%2CF%28-1%2C5%2Bsqrt%2841%29%29%29%2C%0D%0Alocate%28-6.5%2C5-sqrt%2841%29%2B.5%2CF%28-1%2C5%2Bsqrt%2841%29%29%29%0D%0A%0D%0A%0D%0A%0D%0A+++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "Edwin</PRE>\r<BR>\n" );
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