Question
Show that for any angle $\theta$, the equation $x^{2}+y^{2}=r^{2}$ becomes $\left(x^{\prime}\right)^{2}+\left(y^{\prime}\right)^{2}=r^{2}$ when the rotation of axes formulas are applied.
Step 1
Step 1: The given equation is $x^{2}+y^{2}=r^{2}$, which represents a circle with radius $r$ in the $xy$ plane. Show more…
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Key Concepts
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Show that the equation $x^{2}+y^{2}=r^{2}$ is invariant under rotation of axes.
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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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