Write Equations of Rotated Conics in Standard Form
-$$31 x ^ { 2 } + 10 \sqrt { 3 } x y + 21 y ^ { 2 } - 144 = 0$$
A) $\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 9 } = 1$
B) $y ^ { \prime 2 } = - 4 \sqrt { 2 } x ^ { \prime }$
C) $x ^ { + 2 } = - 4 \sqrt { 2 } y ^ { \prime }$
D) $\frac { x ^ { \prime 2 } } { 9 } + \frac { y ^ { \prime 2 } } { 4 } = 1$

Question

Quizplus · Accepted Answer

The given equation is a rotated conic, and to find its standard form, we need to eliminate the xy term. This is done by rotating the coordinate system. The rotation angle $\theta$ can be found using the formula $\tan(2\theta) = \frac{2B}{A-C}$, where $A = 31$, $B = 5\sqrt{3}$ (since the coefficient of xy should be halved), and $C = 21$. Solving for $\theta$ gives us an angle that aligns the conic with the axes in the new $x'$-$y'$ coordinate system. After rotation, the equation transforms into a standard form of an ellipse, which matches option A: $\frac{x'^2}{4} + \frac{y'^2}{9} = 1$, indicating an ellipse with semi-major axis lengths $\sqrt{9}$ and semi-minor axis lengths $\sqrt{4}$, aligned with the $x'$ and $y'$ axes respectively.

Write Equations of Rotated Conics in Standard Form - 31x2+103xy+21y2−144=031 x ^ { 2 } + 10 \sqrt { 3 } x y + 21 y ^ { 2 } - 144 = 031x2+103​xy+21y2−144=0

Write Equations of Rotated Conics in Standard Form
- $31 x ^ { 2 } + 10 \sqrt { 3 } x y + 21 y ^ { 2 } - 144 = 0$