Equation of a Circle

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. You may want to go through
tutorials on finding equations of circles, center, radius and other questions are included in this website.
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 =
-10+10

y-coordinates of Center
k =
-10+10

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.

More References and links to calculators and interactive tutorials

Step by Step Solver Calculator to Find the Points of Intersection of Two Circles.
Step by Step Solver Calculator to Find the Points of Intersection of a Circle and a Line.
Find Equation of a Circle - applet interactive tutorial.
Three Points Circle Calculator.
Match Equations of Circles to Graphs. Excellent interactive activity where equations of circles are matched to graphs.
Find x and y intercepts of Circles - Calculator: A calculator to calculate the x and y intercepts of the graph of a circle given its center and radius.

More References and links analytical tutorials

Step by Step Maths Worksheets Solvers
Find x and y intercepts of Circles - Calculator
Tutorial on Equation of Circle
Tutorial on Equation of Circle (2)
Tutorial on Equation of Circle (3)
Maths Problems and Online Self Tests.