Derivative - Chain Rule, 2

Category: Differential Calculus, Algebra

"Published in Vacaville, California, USA"

Given the following functions:

Find dy/dx.

Solution:

The first thing that we need to do is to get the derivative of the given functions with respect to their independent variables. 

Take the derivative of the first equation with respect to u, we have

Take the derivative of the second equation with respect to x, we have

Since there are three variables in the given functions, then we have to use the Chain Rule in getting dy/dx, we have

Substitute the values of dy/du and du/dx to the above equation, we have

but

then the above equation becomes

Finding Equation - Circle, 13

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Vacaville, California, USA"

Find the equation of a circle that passes through the points of intersection of the circles x² + y² = 2x and x² + y² = 2y, and has its center on the line y = 2. 

Solution:

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

A circle that passes through the points of intersection of two circles and its center at y = 2. (Photo by Math Principles in Everyday Life)

The equation of a chord or radical axis can be solved by subtracting the equations of two circles, as follows

Substitute y = x to either of the equations of a circle, we have

After equating each factor to zero, the values of x are 0 and 1. Hence, the points of intersection of two circles are (0, 0), and (1, 1).

The center of a circle can be solved by using the distance of two points formula as follows

Hence, the center of a circle is C (-1, 2). The radius of a circle is

Therefore, the equation of a circle is

Finding Equation - Circle, 12

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Vacaville, California, USA"

Find the equation of a circle that passes through the points of intersection of the circles x² + y² = 5 and x² + y² - x + y = 4, and through the point (2, -3).

Solution:

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

A circle that passes through the intersection of two circles and a point. (Photo by Math Principles in Everyday Life)

Since the given two circles are non-concentric with their points of intersection, then the equation of another circle can be written as

where k is a constant that represents a family of non-concentric circles. To solve for the value of k, substitute the values of x and y from the given point, we have

Therefore, the equation of a circle is