Write Equations of Rotated Conics in Standard Form
-$$17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$$
A) $\frac { \left( x ^ { \prime } - \frac { 2 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } + \frac { 4 \sqrt { 5 } } { 5 } \right) ^ { 2 } } { 4 } = 1$
B) $x ^ { \prime 2 } = - 16 y ^ { \prime }$
C) $\frac { x ^ { \prime 2 } } { 16 } - \frac { y ^ { \prime 2 } } { 4 } = 1$
D) $\frac { x ^ { \prime 2 } } { 4 } + \frac { y ^ { \prime 2 } } { 16 } = 1$ )

Question

Quizplus · Accepted Answer

The given equation represents a rotated conic section. To find its standard form, we first identify the coefficients of $x^2$, $xy$, and $y^2$, which are 17, -12, and 8, respectively. The rotation angle $\theta$ can be found using the formula $\tan(2\theta) = \frac{B}{A-C}$, where $A$, $B$, and $C$ are the coefficients of $x^2$, $xy$, and $y^2$, respectively. After finding the rotation angle, we apply a rotation of axes transformation to eliminate the $xy$ term, resulting in a conic in its standard form. The correct standard form is given by option A, which represents an ellipse centered at a point shifted from the origin, with semi-major and semi-minor axes lengths that are derived from completing the square and rotating the original equation.

Write Equations of Rotated Conics in Standard Form - 17x2−12xy+8y2−68x+24y−12=017 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 017x2−12xy+8y2−68x+24y−12=0

Write Equations of Rotated Conics in Standard Form
- $17 x ^ { 2 } - 12 x y + 8 y ^ { 2 } - 68 x + 24 y - 12 = 0$