document.write( "Question 549004: sketch a graph at 4x^2-3xy=18 and identify the angle of rotation
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #357862 by Edwin McCravy(20066)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "4x² - 3xy = 18\r\n" );
document.write( " \r\n" );
document.write( "We write it in the form of the general conic:\r\n" );
document.write( " \r\n" );
document.write( "Ax² + Bxy + Cy² + Dx + Ey + F = 0\r\n" );
document.write( " \r\n" );
document.write( "4x² - 3xy + 0y² + 0x + 0y - 18 = 0\r\n" );
document.write( " \r\n" );
document.write( "So A=4, B=-3, C=0, D=0, E=0, F=-18\r\n" );
document.write( " \r\n" );
document.write( "To find the angle to rotate the graph, we use:\r\n" );
document.write( " \r\n" );
document.write( "tan(2Ɵ) = \"B%2F%28A-C%29\" \r\n" );
document.write( " \r\n" );
document.write( "tan(2Ɵ) = \"%28-3%29%2F%284-0%29\"\r\n" );
document.write( " \r\n" );
document.write( "tan(2Ɵ) = \"-3%2F4\"\r\n" );
document.write( " \r\n" );
document.write( "Use the identity for tan(2Ɵ)\r\n" );
document.write( " \r\n" );
document.write( "tan(2Ɵ) = \"2tan%28theta%29%2F%281-%22tan%22%5E2%28theta%29%29\"\r\n" );
document.write( " \r\n" );
document.write( "\"2tan%28theta%29%2F%281-%22tan%22%5E2%28theta%29%29\" = \"-3%2F4\"\r\n" );
document.write( " \r\n" );
document.write( "8tanƟ = -3(1 - tan²Ɵ)\r\n" );
document.write( " \r\n" );
document.write( "8tanƟ = -3 + 3tan²Ɵ\r\n" );
document.write( " \r\n" );
document.write( "-3tan²Ɵ + 8tanƟ + 3 = 0\r\n" );
document.write( " \r\n" );
document.write( "3tan²Ɵ - 8tanƟ - 3 = 0\r\n" );
document.write( " \r\n" );
document.write( "That factors:\r\n" );
document.write( " \r\n" );
document.write( "(tanƟ - 3)(3tanƟ + 1) = 0\r\n" );
document.write( " \r\n" );
document.write( "tanƟ - 3 = 0          3tanƟ + 1 = 0  \r\n" );
document.write( "                         \r\n" );
document.write( "    tanƟ = 3               tanƟ = \"-1%2F3\"\r\n" );
document.write( " \r\n" );
document.write( "       Ɵ = arctan(3)          Ɵ = arctan(\"-1%2F3\")\r\n" );
document.write( "                    \r\n" );
document.write( "   Ɵ = 71.56505118°           Ɵ = 161.5650512° or 341.5650512° \r\n" );
document.write( "       or 251.5650512°\r\n" );
document.write( " \r\n" );
document.write( "Either of those angles will eliminate the xy term when rotated through\r\n" );
document.write( "them.  We choose Ɵ = arctan(3) = 71.56505118°.\r\n" );
document.write( " \r\n" );
document.write( "We draw the triangle:   and use the Pythagorean theorem to find the hypotenuse:  \r\n" );
document.write( " \r\n" );
document.write( " Next we substitute \r\n" );
document.write( "\r\n" );
document.write( "x = x'cosƟ-y'sinƟ = x'\"1%2Fsqrt%2810%29\"-y'\"3%2Fsqrt%2810%29\" = \"1%2Fsqrt%2810%29\"(x' - 3y') \r\n" );
document.write( "y = x'sinƟ+y'cosƟ = x'\"3%2Fsqrt%2810%29\"+y'\"1%2Fsqrt%2810%29\" = \"1%2Fsqrt%2810%29\"(3x' + y') \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "in\r\n" );
document.write( "\r\n" );
document.write( "4x² - 3xy = 18\r\n" );
document.write( "4(\"1%2Fsqrt%2810%29\"(x' - 3y'))² - 3(\"1%2Fsqrt%2810%29\"(x' - 3y'))(\"1%2Fsqrt%2810%29\"(3x' + y')) = 18\r\n" );
document.write( "\r\n" );
document.write( "4(\"1%2Fsqrt%2810%29\")²(x' - 3y')² - 3(\"1%2Fsqrt%2810%29\")²(x' - 3y')(3x' + y') = 18\r\n" );
document.write( "\r\n" );
document.write( "4(\"1%2F10\")(x' - 3y')² - 3(\"1%2F10%29\")(x' - 3y')(3x' + y') = 18\r\n" );
document.write( "\r\n" );
document.write( "Multiply through by 10\r\n" );
document.write( "\r\n" );
document.write( "4(x' - 3y')² - 3(x' - 3y')(3x' + y') = 180\r\n" );
document.write( "\r\n" );
document.write( "4(x'² - 6x'y' + 9y'²) - 3(3x'² - 8x'y' - 3y'²) = 180\r\n" );
document.write( "\r\n" );
document.write( "4x'² - 24x'y' + 36y'² - 9x'² + 24x'y' + 9y'² = 180\r\n" );
document.write( " \r\n" );
document.write( "-5x'² + 45y'² = 180\r\n" );
document.write( "\r\n" );
document.write( "Divide through by 180 to get 1 on the right side\r\n" );
document.write( "\r\n" );
document.write( "\"%28-5%29%2F180\"x'² + \"45%2F180\"y'² = \"180%2F180\"\r\n" );
document.write( "\r\n" );
document.write( "\"%22-x%27%22%5E2%2F36\" + \"%22y%27%22%5E2%2F4\" = 1\r\n" );
document.write( "\r\n" );
document.write( "Write the positive term first:\r\n" );
document.write( "\r\n" );
document.write( "\"%22y%27%22%5E2%2F4\" - \"%22x%27%22%5E2%2F36\" = 1\r\n" );
document.write( "\r\n" );
document.write( "This is a hyperbola of the form \"y%5E2%2Fa%5E2\" - \"x%5E2%2Fb%5E2\" = 1\r\n" );
document.write( "rotated through an angle of Ɵ = arctan(3) = 71.56505118°.\r\n" );
document.write( "a = 2 and b = 6\r\n" );
document.write( "\r\n" );
document.write( "We will draw the x' and y' axes in green:\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "We draw in the defining rectangle, 6 units out the x' axis in\r\n" );
document.write( "both directions, and 2 units out the y' axis in both directions. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "\r\n" );
document.write( "We draw in the asymptotes as the extended diagonals of the\r\n" );
document.write( "defining rectangle, one of which turns out to be the y-axis:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "And we sketch in the hyperbola:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( " Edwin
\n" ); document.write( "
\n" );