The Circle and its Equation
In the following construction you see a circle c with the equation x^{2 }
+ y^{2} = 25, its center M = (0, 0) und the radius r = 5.

Double click on the circle's equation "c: x^{2 }+ y^{2}
= 25" in the left window, change the right hand side of the equation
and press Enter. What happens? Change the right hand side repeatedly and
write down your observations.

Now, try to change the right hand side of the circle's
equation in such a way that the radius becomes a) r =
2, b) r = 4 and c) r= 6. What does the equation look like for an abstract
radius r? Write down your results and conjectures.

Drag the circle with the mouse and keep track of its
equation and its midpoint. Is there any connection between them? Write
down your observations and conjectures.

Use the keyboard to change the circle's equation so that
its center becomes a) M = (4, 2), b) M = (3, 2) and c) M = 2,
1). Write down the resulting equations.

Do you have an idea what the equation of a circle with
abstract center M
= (m, n) and radius r could look like? Write down your conjectures.
Created with
GeoGebra by Markus
Hohenwarter 