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Given the equation of a circle
Find the new equation of a circle if the given circle is rotated counterclockwise about the origin at 45°.
Solution:
To illustrate the given problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
If the given equation is written in rectangular coordinate system, then we need to convert it into polar coordinate system as follows
Since the equation of a circle is only r in polar coordinate system, then there's no way to substitute θ with θ - 45°. This happens if the center of a circle is the origin and it is rotated about its center or origin. Because of this, the new equation of a circle is still the same at any angle. Therefore, the new equation of a circle is
