Determine the appropriate rotation formulas to use so that the new equation contains no xy-term.
-$$4 x ^ { 2 } + 6 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$$
A) $x = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$
B) $x = \frac { 1 } { 2 } u - \frac { \sqrt { 3 } } { 2 } v$ and $y = \frac { \sqrt { 3 } } { 2 } u + \frac { 1 } { 2 } v$
C) $x = - v$ and $y = u$
D) $x = \frac { u } { \sqrt { 2 } } + \frac { v } { \sqrt { 2 } }$ and $y = \frac { u } { \sqrt { 2 } } - \frac { v } { \sqrt { 2 } }$

Question

Quizplus · Accepted Answer

The correct rotation formula to eliminate the $xy$-term in the given equation $4x^2 + 6xy + 4y^2 - 8x + 8y = 0$ is determined by the angle of rotation, which is calculated using the formula $\tan(2\theta) = \frac{B}{A-C}$, where $A$, $B$, and $C$ are coefficients from the general form $Ax^2 + Bxy + Cy^2$. In this case, $A = 4$, $B = 6$, and $C = 4$, leading to $\tan(2\theta) = \frac{6}{4-4} = \text{undefined}$, which suggests a $45^\circ$ rotation or $\frac{\pi}{4}$ radians. The rotation formulas for a $45^\circ$ rotation are $x = \frac{u}{\sqrt{2}} - \frac{v}{\sqrt{2}}$ and $y = \frac{u}{\sqrt{2}} + \frac{v}{\sqrt{2}}$, which corresponds to choice A.

Determine the Appropriate Rotation Formulas to Use So That the New