Equation of a Circle

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This is an HTML5 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

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

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.

Interactive Tutorials

Use an HTML5 Applet for Interactive Exploration

x-coordinates of Center

h =

y-coordinates of Center

k =

Radius of Circle

r =

1 - click on the button above "draw" to start. Use the buttons + and - in the left panel to zoom in and out.

2 - Use the sliders(for chrome and Safari) and/or input boxes 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 and/or input boxes 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
(x - h) 2 + (y - k) 2 = 0

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.
(x - h) 2 + (y - k) 2 = r 2

(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| < 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.

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Updated: February 2015

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