Finding Equation - Circle, 6

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Newark, California, USA"

Find the equation of a circle whose center on the line y = x, tangent to y = 5, and its radius is 2.

Solution:

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

Photo by Math Principles in Everyday Life

In this problem, the center of a circle is unknown and we need to solve it. Since the radius of a circle is given, then we can solve for the coordinates of the center by using the perpendicular distance of a point to the line formula as follows

Substitute the coefficients of the equation of a line that is tangent to a circle, we have

If we choose a negative sign for the radical, then the above equation becomes
 

 

   
Substitute the values of the coordinates of the center of a circle to the above equation, we have
 

 

 

Since the center of a circle is located along the line y = x, then it follows that h = k = 3.

Therefore, the equation of a circle is

 

 

If we choose a positive sign for the radical, then the above equation becomes

Since the center of a circle is located along the line y = x, then it follows that h = k = 7.

Therefore, the equation of a circle is

 

Finding Equation - Circle, 5

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

Find the equation of a circle whose center on the y-axis, and passes through the origin and the point (4, 2).

Solution:

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

Photo by Math Principles in Everyday Life

In this problem, the center of a circle is unknown and we need to solve it. Since the two points of a circle are given, then we can solve for the coordinates of the center of a circle by using the distance of two points formula as follows

Since the given figure is a circle, then it follows that their distances from the center are equal as follows

Square on both sides of the equation, we have

Since the center of a circle is located along the y-axis, then it follows that h = 0.

Hence, the center of a circle is C (0, 5). The radius of a circle can be solved by using the distance of two points formula as follows

Therefore, the equation of a circle is

Graphical Sketch - Circle, 2

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Newark, California, USA"

Write the equation of a circle into standard form and sketch the graph:

Solution:

Consider the given equation above

Since x² and y² have the same coefficients which is 4, then we need to divide both sides of the equation by 4 as follows

Group the left side of the equation by their variables, we have

Complete the square for the two grouped terms, we have

Therefore, the equation of a circle in standard form is

and the center and its radius are

We can now sketch the graph of a circle as follows

Photo by Math Principles in Everyday Life