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.

Rotation of a Parabola

Category: Analytic Geometry

"Published in Vacaville, 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 90°.  

Solution:

To illustrate the given problem, it is better to draw the figure as follows   

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 - 90° 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  

Sketching the Graph of Parabola, 2

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Sketch and label the graph of parabola:

Solution:

Consider the given equation above
   

The given equation is a parabola because only x is the second degree. Let's rewrite the given equation into its standard form as follows

 

Since x is the second degree and y is negative, then the parabola opens downward.

The vertex of a parabola is V(-3, 0).

The value of a which is the distance of a focus from the vertex is 
 

 

The vertex of a focus is F(-3, -3).

The length of a latus rectum in which the midpoint is the focus is 4a = 12.

The distance of one end of a latus rectum to the focus is 2a = 6.

The ends of the latera recta are L(-3, 3) and L'(-3, -9).

Here's the graph of a parabola with labels as follows

Photo by Math Principles in Everyday Life