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Given the equation of a parabola
Find the new equation of a parabola if the given parabola is rotated counterclockwise about the origin at
. 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 |
Since the given angle of rotation is written as inverse tangent function, then we can get the sine and cosine of the given angle of rotation by using basic trigonometric functions of a right triangle.
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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
Next, substitute θ with θ - ϕ and then expand using the sum and difference of two angles formula, we have
Convert the above equation into rectangular coordinate system in order to get its final equation. Therefore, the new equation of a parabola is
Note: If the axes of any conic sections are not parallel to x and y axes, then the equation of any conic sections has xy term which is the general equation of any conic sections like parabola, ellipse, and hyperbola. Circle has no xy term always in any cases.