Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term.
-$$x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$$
A) $x = \frac { \sqrt { 2 } } { 2 } \left( x ^ { \prime } - y ^ { \prime } \right) ; y = \frac { \sqrt { 2 } } { 2 } \left( x ^ { \prime } + y ^ { \prime } \right)$
B) $x = - y ^ { \prime } ; y = x ^ { \prime }$
C) $x = \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 } x ^ { \prime } - \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 } y ^ { \prime } ; y = \frac { \sqrt { 2 - \sqrt { 2 } } } { 2 } x ^ { \prime } + \frac { \sqrt { 2 + \sqrt { 2 } } } { 2 } y ^ { \prime }$
D) $x = \frac { 1 } { 2 } x ^ { \prime } - \frac { \sqrt { 3 } } { 2 } y ^ { \prime } ; y = \frac { \sqrt { 3 } } { 2 } x ^ { \prime } + \frac { 1 } { 2 } y ^ { \prime }$

Question

Quizplus · Accepted Answer

The rotation formula that eliminates the $x'y'$-term for the given equation is derived by setting the coefficient of the $x'y'$ term to zero in the rotated system. This is achieved by choosing a rotation angle $\theta$ that satisfies $\tan(2\theta) = \frac{2B}{A-C}$, where $A$, $B$, and $C$ are the coefficients of $x^2$, $xy$, and $y^2$ respectively. For the given equation, $A = C = 1$ and $B = 2$, leading to an angle of $\theta = 45^\circ$. The rotation formulas for a $45^\circ$ rotation are given by option A.

Write the Appropriate Rotation Formulas So That in a Rotated x2+2xy+y2−8x+8y=0x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0x2+2xy+y2−8x+8y=0

Write the Appropriate Rotation Formulas So That in a Rotated $x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0$