Rotate the axes to eliminate the $x y$-term in the following equation.
$$18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$$
A) $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$
B) $6 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } + 6 \sqrt { 3 } x ^ { \prime } = 0$
C) $6 \left( x ^ { \prime } \right) ^ { 2 } - 6 y ^ { \prime } = 0$
D) $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { x ^ { \prime } } { 6 } = 0$
E) $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } + \frac { y ^ { \prime } } { 12 } = 0$

Question

Quizplus · Accepted Answer

The equation can be transformed by rotating the axes to eliminate the $xy$-term. The angle of rotation $\theta$ that accomplishes this is given by $\tan(2\theta) = \frac{B}{A-C}$, where $A$, $B$, and $C$ are the coefficients of $x^2$, $xy$, and $y^2$ respectively in the given equation. Here, $A = 18$, $B = -12\sqrt{3}$, and $C = 6$. Substituting these values gives $\tan(2\theta) = \frac{-12\sqrt{3}}{18-6} = -2\sqrt{3}$, which corresponds to an angle $\theta = 30^\circ$ or $\theta = -30^\circ$. Rotating the axes by this angle eliminates the $xy$-term and simplifies the equation to a form involving only $x'$ and $y'$, which corresponds to choice A.

Rotate the Axes to Eliminate The xyx yxy -Term in the Following Equation

Rotate the Axes to Eliminate The $x y$ -Term in the Following Equation