Question

The equation of a circle centered at the origin with a radius of unit length is x2 + y2 = 1. This equation changes if the center of the circle is not located at the origin or the radius is not of unit length. You will use the GeoGebra geometry tool to examine how the equation of a circle changes as you move the center and change the radius. Go to equation of a circle , and complete each step below. If you need help, follow these instructions for using GeoGebra.How does the equation change when the radius changes? Unlike h and k, why is r always positive?

Answer 1

Given the unitary circle equation centered at the origin:

`x^2+y^2=1`

If the circle is not centered at the origin, but at the point (h, k), the equation becomes:

`(x-h)^2+(y-k)^2=1`

When h = k = 0, we have the particular case of a circle centered at the origin. Now, if the circle is not unitary, the equation becomes:

`x^2+y^2=r^2`

So when the radius is 1 (unitary circle), we have our initial case. Combining these results, the general equation of a circle of radius r and centered at (h, k) is:

`(x-h)^2+(y-k)^2=r^2`

r is always positive because it represents the measure of the radius length, and the length is always positive. On the other hand, the equation always represents a positive value, because the square of any number is always positive (or zero).