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Thursday, November 13, 2014

Rotation of a Parabola, 2

Category: Analytic Geometry

"Published in Newark, California, USA"

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

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. 


 
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.