Rotation of a Line

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 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

Next, substitute θ with θ - 45° 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

 

Solving Radical Equations, 4

Category: Algebra

"Published in Newark, California, USA"

Find the roots of the equation for

Solution:

The first thing that we need to do is to examine the given equation first if we can simplify or not. In this case, √3y + 4 is a common factor on both sides of the equation. Cancel their common factor, we have

Since y² will be eliminated from the above equation, then it becomes a linear equation and we can solve for the value of y. The value of y is

 

 

 

Next, we need to check if the value of y is a root or not since the given equation is a radical equation. There are cases that the roots are extraneous. Let's check the value of y from the original equation, we have
 

 

 

 

Since both sides of the equation are equal, therefore, y = 6 is the root of the equation.

Finding the Equation of Ellipse, 2

Category: Analytic Geometry

"Published in Vacaville, California, USA"

Find the equation of ellipse with center C(0, 4), foci F₁(0, 0) and F₂(0, 8), and major axis of length 10 units.

Solution:

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

Photo by Math Principles in Everyday Life

If the coordinates of the center and foci and the length of major axis are given, then c = 4, and a = 10/2 = 5. The value of b is

Since the coordinates of the foci have the same x-coordinate, then the major axis is parallel to y-axis. Therefore, the equation of ellipse whose center is C(h, k) in standard form is
 

 

 

In general form, the equation of ellipse is