This is an applet that generates two graphs of circles. The equations of these circles are of the form:
^{2} + (y - k)^{2} = r^{2}
You can control the parameters of the blue circle by changing parameters h, k and r. The second circle is the red one and it is generated randomly. As an exercise, you need to find an equation to the red circle. We suggest that you first use an analytical method to find the equation of the circle and then use the applet to change h, k and r to solve the same question graphically. Finally compare the two results. This exercise helps you in problem solving and also to gain a deep understanding of the properties of the circle. TUTORIAL
1 - click on the button above "click here to start" and MAXIMIZE the window obtained.
2 - From the graph, determine the x and y coordinates of the center of the circle (red point inside the circle) and a point on the graph and use an analytical method to find an equation of the form ^{2} + (y - k)^{2} = r^{2}
where h and k are the x and y coordinates of the center and r is the radius of the circle. You may use the method in example 5 below. 3 - Use the sliders to change h, k and r (top left) so that the two graphs are the same. Read the values of h, k and r and compare these values to those found analytically above. 4 - Generate another question by clicking on the button "new parabola" (bottom left) . You can generate as many questions as you wish. 5 - Example: A circle has center at (0,4) and passes through the point (3,0). Find an equation to this circle of the form (x - h) ^{2} + (y - k)^{2} = r^{2}.
6 - Solution to the example in 5. The x and y coordinates of the center gives the values of h and k respectively. Hence h = 0 and k = 4. The equation can be written as x ^{2} + (y - 4)^{2} = r^{2}. r is the distance between the center of the circle and any point on the circle.
r = sqrt((3 - 0) ^{2} + (0 - 4)^{2}) = 5
The equation of the circle can be written as x ^{2} + (y - 4)^{2} = 25.
You can check that point (3,0) is on the graph of the circle: 3 ^{2} + (0 - 4)^{2} = 9 + 16 = 25.
tutorial on circles |