This is an applet to explore the equation of a circle and the properties of the circle. The equation used is the standard equation that has the form

where h and k are the x- and y-coordinates of the center of the circle and r is the radius.
The exploration is carried out by changing the parameters h, k and r included in the above equation. Follow the steps in the tutorial below. If you wish to go through a tutorial on finding equations of circles, center, radius and other questions Go here.

Similar tutorials on the ellipse , parabola and the hyperbola can be found in this site.

TUTORIAL

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1 - click on the button above "click here to start" and MAXIMIZE the window obtained.

2 - Use the sliders to set parameters h and k to zero and parameter r to 1. Check that the circle shown has the center at (0,0) and radius equal to 1.

3 - Special case: Use the sliders to set r to zero and parameters h and k to different values, the graph of the circle is a point, Explain.(Hint:Solve the equation

4 - Keep r equal to 1 and shift the circle by changing h and k. Check that the center of the circle is at (h , k).

5 - Keep h and k constant and change r. Check that the circle has radius r.

6 - Set h, k and r to 1. The circle has one point of intersection with the x-axis and one point of intersection with the y-axis. These are called the x and y intercepts. Find these points analytically using the equation of the circle.

(Hint: To find the x-intercepts set y = 0 in the equation and solve for x. To find the y-intercepts set x = 0 in the equation and solve for y.)

7- Set r to 2 and h to a certain value. Change k from -1.8 to 1.8 (|h| less than r). How many x-intercepts are there? Set k to 2 (the radius), How many x-intercepts are there? Set k to -2, how many x-intercepts are there? Set k to values greater than 2 (the radius), how many x-intercepts are there? Set k to values smaller than -2, how many x-intercepts are there? Explain analytically.

8- Try the same exploration as in 7 above with the y-intercepts by changing the value of h.

9- Exercise: Find (analytically) values of h, k and r such that the circle associated with these values has no x or y-intercepts. Check your answer graphically.