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

Rotation of a Circle, 2

Category: Analytic Geometry

"Published in Newark, California, USA"

Given the equation of a circle

Find the new equation of a circle if the given circle is rotated counterclockwise about the origin at 30°.  

Solution:

The first thing that we need to do is to write the given equation in standard form as follows

From the standard form, we can draw the circle 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 - 30° 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 circle is    

 

Rotation of a Circle

Category: Analytic Geometry

"Published in Newark, California, USA"

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  

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

 

Rotation of a Line, 3

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Given the equation of a line

Find the new equation of a line if the given line is rotated counterclockwise about the origin at 60°.  

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 - 60° 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 line is