document.write( "Question 830830: Solve Oblique conic : 10x2 +24xy + 17 y2 -9 = 0
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Algebra.Com's Answer #501047 by Edwin McCravy(20056) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Since the other tutor gave you such a pitiful answer, I thought\r\n" );
document.write( "I'd take the trouble to give you what you wanted.  It's quite a\r\n" );
document.write( "tough problem.\r\n" );
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document.write( "The general form of a conic is \r\n" );
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document.write( "A=10, B=24, C=17\r\n" );
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document.write( "Discriminant B²-4AC = 24²-4(10)(17) = -104\r\n" );
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document.write( "This being negative tell us that the equation represents an\r\n" );
document.write( "ellipse.\r\n" );
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document.write( "We will rotate the axes through an angle of  where\r\n" );
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document.write( "Let , then\r\n" );
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document.write( "Divide both sides by 2\r\n" );
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document.write( ";  \r\n" );
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document.write( "We can choose either angle.  I'll choose the positive one.\r\n" );
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document.write( "That's a 3-4-5 right triangle   \r\n" );
document.write( "so ,\r\n" );
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document.write( "We make the substitution \r\n" );
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document.write( "or\r\n" );
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document.write( "[Most books use x' and y', but I am using capital X and Y because\r\n" );
document.write( "primed letters are difficult to work with.  So be careful in\r\n" );
document.write( "the rest of the problem to distinguish between capital X and small x\r\n" );
document.write( "and capital Y and small y.  Remember the capital letters refer to the\r\n" );
document.write( "rotated axes and small ones refer to the original un-rotated axes.] \r\n" );
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document.write( "Substitute into\r\n" );
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document.write( "Multiply through by 25\r\n" );
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document.write( "10(3X-4Y)²+24(3X-4Y)(3Y+4X)+17(3Y+4X)²-225 = 0\r\n" );
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document.write( "10(9X²-24XY+16Y²)+24(9XY+12X²-12Y²-16XY)+17(9Y²+24XY+16X²)-225 = 0\r\n" );
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document.write( "90X²-240XY+160Y²+24(12X²-7XY-12Y²)+153Y²+408XY+272X²-225 = 0\r\n" );
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document.write( "90X²-240XY+160Y²+288X²-168XY-288Y²+153Y²+408XY+272X²-225 = 0\r\n" );
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document.write( "650X²+25Y²-225=0\r\n" );
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document.write( "Divide through by 25\r\n" );
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document.write( "26X²+Y²-9 = 0\r\n" );
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document.write( "To get that in standard form,\r\n" );
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document.write( "26X²+Y² = 9\r\n" );
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document.write( "Divide through by 9 to get 1 on the right\r\n" );
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document.write( "Divide numerator and denominator of first term by 26\r\n" );
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document.write( "The center is the origin, because\r\n" );
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document.write( "can be written:\r\n" );
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document.write( ", and is of the form\r\n" );
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document.write( " since 9=a²>b²=, and\r\n" );
document.write( "where (h,k) is the center.\r\n" );
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document.write( "center = (X,Y) = (x,y) = (0,0)\r\n" );
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document.write( "a²=9, so a=3, so the (X,Y) coordinates of the vertex is on the Y-axis\r\n" );
document.write( "is (X,Y) = (0,3).  We translate this to its (x,y) coordinates, by using\r\n" );
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document.write( "The upper left vertex is (x,y) = (,)\r\n" );
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document.write( "By symmetry, the lower right vertex is (,)\r\n" );
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document.write( "To find the foci, we need the value c,\r\n" );
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document.write( "c² = a²-b²\r\n" );
document.write( "c² = 9-\r\n" );
document.write( "c² = \r\n" );
document.write( "c² = \r\n" );
document.write( " c = \r\n" );
document.write( " c = \r\n" );
document.write( " c = \r\n" );
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document.write( "So the foci on the Y-axis are (X,Y) = (0,)\r\n" );
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document.write( "We translate the one with the positive Y to its (x,y) coordinates, by using\r\n" );
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document.write( "The upper left focus is (x,y) = (,)\r\n" );
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document.write( "By symmetry, the lower right focus is (,)\r\n" );
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document.write( "To find the domain and range exactly is really murder.\r\n" );
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document.write( "To find the range exactly\r\n" );
document.write( "1. Solve the original equation for y using the quadratic formula\r\n" );
document.write( "2. You will have two functions, one using + and one using -\r\n" );
document.write( "3. Find their derivatives\r\n" );
document.write( "4. Set each equal to 0 and solve for x\r\n" );
document.write( "5. Substitute in the result of 1 to find y in each\r\n" );
document.write( "6. The range will be [smaller value of y, larger value of y]\r\n" );
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document.write( "To find the domain exactly\r\n" );
document.write( "1. Interchange x and y in the original equation.\r\n" );
document.write( "2. Solve the that equation for y using the quadratic formula\r\n" );
document.write( "3. You will have two functions, one using + and one using -\r\n" );
document.write( "4. Find their derivatives.\r\n" );
document.write( "5. Set each equal to 0 and solve for x\r\n" );
document.write( "6. Substitute in the result of 2 to find y in each\r\n" );
document.write( "7. The domain will be [smaller value of y, larger value of y]   \r\n" );
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document.write( "That'll take many hours.  I just used a TI-84 to find the \r\n" );
document.write( "approximate domain and range,\r\n" );
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document.write( "The approximate domain is [-2.47386337,2.47386337]\r\n" );
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document.write( "The approximate range is [-2.283567,2.283567]\r\n" );
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document.write( "Edwin
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