How To Write The Equation of A Circle

Equation Of A Circle

A circle is a cross sectional cut of a conic section, just as is a parabola, ellipse and hyperbola.  A circle is characterized by its uniform radius and symmetry about the x and y axes.   The standard equation of a circle is written in the following form.

 

(x – h)2 + (y – k)2 = r2

 

The x and y coordinates represent any point that lies along the curve of the circle, h and k are the x and y coordinates, respectively of the center point of the circle and r is the radius of the circle.  The circle is located at the intersection of the x and y axis (the origin), when h = k = 0 and at this point the equation becomes, x2 + y2 = r2.  Additionally, this equation can be rewritten as r = (x2 + y2)1/2.

 

Let’s apply this concept to a few examples.

 

1.) Find the coordinates of a possible point that lies along the curve of the circle, based on the following equation.

 

(x – 4)2 + (y – 6)2 = 64

 

We can see that the center of the circle is located at x = 4 and y = 6.  We also know that

r2 = 64 and so r = 8.  Since the radius is 8, we select a simple point       to consider, 8 units to the right of the center point.  This means that we are considering the point (4 + 8, 6 + 0) or (12, 6) on the curve.  We can check make          sure this point correctly lies along the curve by plugging in these values for the x   and y coordinates and making sure our equation is equivalent to 64.

 

At (12, 6) we have:

 

(x – 4)2 + (y – 6)2 = 64

(12 – 4)2 + (6 – 6)2 = 64

(8)2 + (0)2 = 64

(8)2 = 64, Correct.

 

2.) What is the radius of a circle centered at (2, 3) and has the point (-6, 9) lying on its curve.

 

The equation of our circle has the form, (x – h)2 + (y – k)2 = r2.  And if we apply the values for the center point, we have (x – 2)2 + (y – 3)2 = r2.  Now we add the values for the point lying along the curve giving us (-6 – 2)2 + (9 – 3)2 = r2.  From here we can determine the radius of the circle.

 

(-6 – 2)2 + (9 – 3)2 = r2

(-8)2 + (6)2 = r2

64 + 36 = r2

100 = r2

r = 10

 

The radius of this circle is 10.

 

 

REFERENCE:

Bittinger, Marvin L., Ellenbogen, David, Keedy, Mervin L. (1994).  Intermediate Algebra: Concepts and Application (4th ed.).  USA: Addison-Wesley Publishing Company, Inc.