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The simplest example of a second-degree equation involving a cross term is $xy=1.$ This equation can be solved for y to obtain $y=\frac{1}{x}.$ The graph of this function is called a rectangular hyperbola as shown.
The asymptotes of this hyperbola are the x and y coordinate axes. To determine the angle $\theta $ of rotation of the conic section, we use the formula $\text{cot}\phantom{\rule{0.2em}{0ex}}2\theta =\frac{A-C}{B}.$ In this case $A=C=0$ and $B=1,$ so $\text{cot}\phantom{\rule{0.2em}{0ex}}2\theta =(0-0)\text{/}1=0$ and $\theta =45\text{\xb0}.$ The method for graphing a conic section with rotated axes involves determining the coefficients of the conic in the rotated coordinate system. The new coefficients are labeled ${A}^{\prime},{B}^{\prime},{C}^{\prime},{D}^{\prime},{E}^{\prime},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{F}^{\prime},$ and are given by the formulas
The procedure for graphing a rotated conic is the following:
Identify the conic and calculate the angle of rotation of axes for the curve described by the equation
In this equation, $A=13,B=\mathrm{-6}\sqrt{3},C=7,D=0,E=0,$ and $F=\mathrm{-256}.$ The discriminant of this equation is $4AC-{B}^{2}=4\left(13\right)\left(7\right)-{\left(\mathrm{-6}\sqrt{3}\right)}^{2}=364-108=256.$ Therefore this conic is an ellipse. To calculate the angle of rotation of the axes, use $\text{cot}\phantom{\rule{0.2em}{0ex}}2\theta =\frac{A-C}{B}.$ This gives
Therefore $2\theta ={120}^{\text{o}}$ and $\theta ={60}^{\text{o}},$ which is the angle of the rotation of the axes.
To determine the rotated coefficients, use the formulas given above:
The equation of the conic in the rotated coordinate system becomes
A graph of this conic section appears as follows.
Identify the conic and calculate the angle of rotation of axes for the curve described by the equation
The conic is a hyperbola and the angle of rotation of the axes is $\theta =22.5\text{\xb0}.$
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