document.write( "Question 553901: Please help me, I don't understand conic intersection at all.
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Algebra.Com's Answer #361115 by AnlytcPhil(1810)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This problem has no real solutions. Let me explain what is \r\n" );
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document.write( "The equation \r\n" );
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document.write( "x² - 4y² = 16\r\n" );
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document.write( "if you were to solve it for y and plot a bunch of points, like these:\r\n" );
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document.write( "(-8.5,-3.75), (-8.5,3.75), (-5,-1.5), (-5,1.5), (-4,0)\r\n" );
document.write( "(4,0), (5,-1.5), (5,1.5), (8.5,-3.75), (8.5,3.75)\r\n" );
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document.write( "you would get this graph, called a hyperbola.  It is called a \"conic section\"\r\n" );
document.write( "because there are several kinds of curves that can be gotten by slicing a\r\n" );
document.write( "cone different ways.  The other tutor is right that you need not be\r\n" );
document.write( "concered about that at this time.  Anyway here is the graph of x² - 4y² = 16.\r\n" );
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document.write( "Now let's look at the other equation, \r\n" );
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document.write( "y = 3x - 3\r\n" );
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document.write( "If you were to plot some points, like these:\r\n" );
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document.write( "(0,-3), (1,0), (2,3)\r\n" );
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document.write( "you will get a straight line like thisL\r\n" );
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document.write( "But if you put them both on the same graph, like this:\r\n" );
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document.write( "you find that they don't intersect at all!  That means there\r\n" );
document.write( "cannot be any REAL point where they intersect. \r\n" );
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document.write( "Suppose you did as the other tutor said:\r\n" );
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document.write( "x² - 4y² = 16\r\n" );
document.write( "y = 3x - 3\r\n" );
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document.write( "Substitute 3x - 3 for y in the first equation:\r\n" );
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document.write( "x² - 4(3x - 3)² = 16\r\n" );
document.write( "x² - 4(3x - 3)(3x - 3) = 16\r\n" );
document.write( "x² - 4(9x² - 18x + 9) = 16\r\n" );
document.write( "x² - 36x² + 72x - 36 = 16\r\n" );
document.write( "-35x² + 72x - 36 = 16\r\n" );
document.write( "-35x² + 72x - 52 = 0\r\n" );
document.write( " 35x² - 72x + 52 = 0\r\n" );
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document.write( "That does not factor and if you use the quadratic formula:\r\n" );
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document.write( "x = \"%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"  \r\n" );
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document.write( "x =  \r\n" );
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document.write( "x = \"%2872+%2B-+sqrt%285184-7280+%29%29%2F%2870%29+\" \r\n" );
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document.write( "x = \"%2872+%2B-+sqrt%28-2096%29%29%2F70+\"\r\n" );
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document.write( "x = \"%2872+%2B-+i%2Asqrt%282096%29%29%2F70+\"\r\n" );
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document.write( "x = \"%2872+%2B-+i%2Asqrt%2816%2A131%29%29%2F70+\"\r\n" );
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document.write( "x = \"%2872+%2B-+4i%2Asqrt%28131%29%29%2F70+\"\r\n" );
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document.write( "x = \"%284%2818+%2B-+i%2Asqrt%28131%29%29%29%2F70+\"\r\n" );
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document.write( "x = \"%282%2818+%2B-+i%2Asqrt%28131%29%29%29%2F35+\"\r\n" );
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document.write( "Which is an imaginary answer, not a real answer\r\n" );
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document.write( "Now if the second equation had been 10y + 3x = 0\r\n" );
document.write( "instead and your problem had been:\r\n" );
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document.write( "x² - 4y² = 16\r\n" );
document.write( "10y + 3x = 0\r\n" );
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document.write( "and we found and plotted these points, say\r\n" );
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document.write( "(0,0), (4, -1.2), (7,-2.1) \r\n" );
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document.write( "and had gotten this line instead, there would have\r\n" );
document.write( "been two real solutions at the points where these two\r\n" );
document.write( "graphs intersect, as shown with the two small circles. \r\n" );
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document.write( "Suppose you did as the other tutor said in this\r\n" );
document.write( "problem:\r\n" );
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document.write( "x² - 4y² = 16\r\n" );
document.write( "10y = 3x\r\n" );
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document.write( "Solve the second equation for y\r\n" );
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document.write( "y = \"3x%2F10\"\r\n" );
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document.write( "Substitute \"3x%2F10\" for y in the first equation:\r\n" );
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document.write( "x² - 4y² = 16\r\n" );
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document.write( "x² - 4\"%283x%2F10%29%5E2\" = 16\r\n" );
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document.write( "x² - 4\"%289x%5E2%2F100%29\" = 16\r\n" );
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document.write( "x² - \"%289x%5E2%2F25%29\" = 16\r\n" );
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document.write( "Multiply through by 25 to clear of fractions:\r\n" );
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document.write( "25x² - 9x² = 400\r\n" );
document.write( "      16x² = 400\r\n" );
document.write( "        x² = 25\r\n" );
document.write( "         x = ±5\r\n" );
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document.write( "Using x = +5 you would have gotten\r\n" );
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document.write( "y = \"3x%2F10\"\r\n" );
document.write( "y = \"3%285%29%2F10%7D%7D%0D%0Ay+=+%7B%7B%7B15%2F10\"\r\n" );
document.write( "y = \"3%2F2\"\r\n" );
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document.write( "So one solution would be the point (5,\"3%2F2\") or (5,1.5)\r\n" );
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document.write( "Using x = -5 you would have gotten\r\n" );
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document.write( "y = \"3x%2F10\"\r\n" );
document.write( "y = \"3%28-5%29%2F10%7D%7D%0D%0Ay+=+%7B%7B%7B-15%2F10\"\r\n" );
document.write( "y = \"-3%2F2\"\r\n" );
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document.write( "So the other solution would be (-5,\"-3%2F2\") or (-5,-1.5)\r\n" );
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document.write( "Notice that the points where the graphs intersect are indeed\r\n" );
document.write( "those very points (5,1.5) and (-5,-1.5)\r\n" );
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document.write( "Edwin
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