Write the Appropriate Rotation Formulas So That in a Rotated $$4 x ^ { 2 } + 5 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$$

question 105

Multiple Choice

Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term.
- $$4 x ^ { 2 } + 5 x y + 4 y ^ { 2 } - 8 x + 8 y = 0$$

• 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) \(x = \frac { 1 } { 2 } x ^ { \prime } - \frac { \sqrt { 3 } } { 2 } y ^ { \prime } ; y = \frac { \sqrt { 3 } } { 2 } x ^ { \prime } + \frac { 1 } { 2 } y ^ { \prime }\)

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Definitions:

Fine Motor Skills

Fine motor skills involve the coordination of small muscles in movements—usually involving the synchronization of hands and fingers—with the eyes.

Small Body Movements

Minor physical actions or gestures, often involuntary, that can be indicative of underlying emotions or intentions.

Motor Skills

The abilities related to the movement and coordination of the body's muscles, including both fine motor skills (small movements, such as writing) and gross motor skills (larger movements, such as walking).

Fingers

The digits on the human hand, which are essential for gripping and manipulating objects, consisting of the thumb, index, middle, ring, and little finger.