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

Finding Equation - Circle, 11

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Vacaville, California, USA"

Find the equation of a circle if it is inscribed in the triangle determined by the lines y = 0, 3x - 4y + 30 = 0, and 4x + 3y = 60.

Solution:

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

A circle that is inscribed in a triangle from the three lines. (Photo by Math Principles in Everyday Life)

The first thing that we need to do is to get the equations of the angle bisectors of the given lines in order to get the coordinates of the center of a circle as well as its radius. 

A circle that is inscribed in a triangle from the three lines. (Photo by Math Principles in Everyday Life)

By using the distance of a point to the line formula, the angle bisector of lines y = 0 and 3x - 4y + 30 = 0 will be equal to

By using the distance of a point to the line formula, the angle bisector of lines y = 0 and 4x + 3y = 60 will be equal to

Hence, the angle bisectors are

Two angle bisectors are enough to use in order to get their intersection which is the center of a circle. Subtract the second equation from the first equation, we have

Substitute the value of y to either of the two equations above, we have

  
The coordinates of the center of a circle is C (5, 5). Since the center of a circle is located above the side of a triangle which is also the y-axis, then the radius of a circle is equal to 5. 

Therefore, the equation of a circle is