Rotating a Circle Show that the equation x ^ 2 + y ^ 2 = r ^ 2 is invariant under rotation

step 1 We're asked to answer a question about rotating a circle using the method of rotation of conics.

Rotating a Circle Show that the equation x ^ 2 + y ^ 2 =...

Key Concepts

Invariance Under Rotation

Invariance under rotation means that the form of a geometric equation remains unchanged when the coordinate system is rotated. For example, the equation of a circle (x² + y² = r²) is invariant under rotation because the distance from any point on the circle to the center remains the same regardless of how the axes are oriented. This concept is essential in understanding the symmetry and conservation properties of geometric figures.

Rotation of Axes

Rotation of axes refers to changing the coordinate system by rotating the coordinate lines by a fixed angle while keeping the origin fixed. This process involves transforming the original coordinates (x, y) into new coordinates (x', y') using specific formulas that incorporate sine and cosine functions of the rotation angle, thereby reorienting the coordinate system without altering the underlying geometry.

Orthogonal Transformations

Orthogonal transformations, such as rotations, are represented by orthogonal matrices whose columns (or rows) form an orthonormal basis. These transformations preserve distances and angles, making them fundamental in geometry and linear algebra. When applied to coordinates, the structure of geometric figures is maintained, which is why the properties of shapes, like a circle, remain invariant under rotation.

Trigonometric Identities

Trigonometric identities, particularly the identity cos²? + sin²? = 1, play a crucial role when transforming coordinates under rotation. These identities simplify the algebraic manipulation of transformed equations and ensure that terms combine appropriately to confirm the invariance of a figure (for instance, showing that x'² + y'² = x² + y²).

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