Let an x^' y^' -coordinate system be obtained by rotating an...

Let an $x^{\prime} y^{\prime}$ -coordinate system be obtained by rotating an $x y$ -coordinate system through an angle $\theta .$ Prove: For every value of $\theta,$ the equation $x^{2}+y^{2}=r^{2}$ becomes the equation $x^{\prime 2}+y^{\prime 2}=r^{2} .$ Give a geometric explanation.

Let an $x^{\prime} y^{\prime}$ -coordinate system be obtained by rotating an
$x y$ -coordinate system through an angle $\theta .$ Prove: For every value of $\theta,$ the equation $x^{2}+y^{2}=r^{2}$ becomes the equation $x^{\prime 2}+y^{\prime 2}=r^{2} .$ Give a geometric explanation.

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Key Concepts

Coordinate Rotation

This concept involves transforming a coordinate system by rotating the axes by a given angle. The new coordinates are determined by linear combinations of the original coordinates using sine and cosine functions. Such a transformation preserves the relative positioning of points while changing their coordinate representations.

Distance Preservation Under Rotation

Rotation is an isometry, meaning it preserves the distances between points. Since the distance from the origin to any point (measured by the square root of the sum of the squares of the coordinate values) remains the same, any geometric figure defined by a fixed distance, such as a circle, retains its shape and size after rotation.

Invariance of the Circle

A circle is defined as the set of points that are equidistant from a central point. Because rotations preserve distances, the equation representing a circle (the sum of the squares of the coordinates being equal to the radius squared) stays unchanged after the coordinate system is rotated. This invariance explains why the circle's equation remains identical in the rotated coordinate system.

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