Use axis of rotation formulas for x and y to transform the quadratic equation to an equation in (u, v) coordinates with no
cross-product term.
-$7 x ^ { 2 } + 6 \sqrt { 3 } x y + 13 y ^ { 2 } = 57$
A) $\frac { 16 u ^ { 2 } } { 57 } + \frac { 4 v ^ { 2 } } { 57 } = 1$
B) $\mathrm { v } = \frac { \mathrm { u } ^ { 2 } } { 4 }$
C) $\frac { 4 u ^ { 2 } } { 57 } + \frac { 16 v ^ { 2 } } { 57 } = 1$
D) $\frac { 3 \mathrm { u } ^ { 2 } } { 114 } + \frac { \mathrm { v } ^ { 2 } } { 114 } = 1$

Question

Quizplus · Accepted Answer

The given quadratic equation is $7x^2 + 6\sqrt{3}xy + 13y^2 = 57$. To eliminate the cross-product term $xy$, we use the formula for the axis of rotation, which involves finding the angle $\theta$ that satisfies $\tan(2\theta) = \frac{2B}{A-C}$, where $A$, $B$, and $C$ are the coefficients from the general form $Ax^2 + 2Bxy + Cy^2$. Here, $A = 7$, $B = 3\sqrt{3}$, and $C = 13$. Solving for $\theta$ gives us the angle of rotation. After applying the rotation, the equation transforms into a form without the $xy$ term, which can be further simplified into the standard form of an ellipse, $\frac{u^2}{a^2} + \frac{v^2}{b^2} = 1$, where $u$ and $v$ are the coordinates in the rotated system. The correct transformation, based on the coefficients and the process of eliminating the $xy$ term, leads to the equation in choice A.

Use Axis of Rotation Formulas for X and Y to Transform