Use Rotation of Axes Formulas
-$$x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0 ; \quad \theta = 45 ^ { \circ }$$
A) $x ^ { \prime 2 } = - 4 \sqrt { 2 } y ^ { \prime }$
B) $3 x ^ { \prime 2 } - 4 \sqrt { 2 } x ^ { \prime } y ^ { \prime } + y ^ { \prime 2 } = 0$
C) $x ^ { \prime 2 } = - 4 \sqrt { 2 } y ^ { \prime 2 }$
D) $2 x ^ { \prime 2 } - \sqrt { 2 } x ^ { \prime } y ^ { \prime } + 2 y ^ { \prime 2 } = 0$

Question

Quizplus · Accepted Answer

To rotate the given equation by $45^\circ$, we use the rotation of axes formulas: $x = x' \cos \theta - y' \sin \theta$ and $y = x' \sin \theta + y' \cos \theta$, where $\theta = 45^\circ$, $\cos 45^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2}$. Substituting these into the given equation and simplifying leads to the equation in choice A, $x'^2 = -4\sqrt{2}y'$.

Use Rotation of Axes Formulas - x2+2xy+y2−8x+8y=0;θ=45∘x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0 ; \quad \theta = 45 ^ { \circ }x2+2xy+y2−8x+8y=0;θ=45∘

Use Rotation of Axes Formulas
- $x ^ { 2 } + 2 x y + y ^ { 2 } - 8 x + 8 y = 0 ; \quad \theta = 45 ^ { \circ }$