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This is an applet to explore the properties of the ellipse given by the following equation:

(x - h)2 / a2 + (y - k)2 / b2 = 1

where h, k, a and b are real numbers. a and b are positive.
The exploration is carried out by changing the parameters h, k, a and b. Follow the steps in the tutorial below. Another tutorial on ellipses are included in this website.
Similar tutorials on circle , parabola and the hyperbola can be found in this site.
Interactive Tutorial
Explore ellipses using the HTML5 applet below
h =
k =
a = b =

HTML5 Applet to Explore Equations of Ellipses
  1. When you first open this page,h = 0, k = 1, a = 3 and b = 1. The graph of the equation is a shifted ellipse.
  2. Set h = k = 0 and b = 1, change a to 2. The line segment formed by the x-intercepts is called the major axis. The line segment formed by the y-intercepts is called the minor axis. Check that the length of the major axis is equal to 2a and that of the minor axis is equal to 2b. Explain analytically (Hint: find the x and y-intercepts and the distance between the segments defined above).
  3. Change h and k. What happens to the ellipse? Explain analytically. The center of the ellipse is the point of intersection of the axes defined above. Change h and k and see that the center has coordinates (h,k).
  4. Set h and a to the same value, 2 for example. The graph of the ellipse is always (when k and b change) tangent to the y-axis. Explain analytically.
  5. Set k and b to the same value, 1.6 for example. The graph of the ellipse is always (when h and a change) tangent to the x-axis. Explain analytically.
  6. Set h, k a and b to some values so that the graph has 2 x-intercepts and 2 y-intercepts. Approximate the coordinates of the x and y intercepts graphically. Find the x and y intercepts analytically and compare the two results.
    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 ellipse associated with these values has no x or y-intercepts. Check your answer graphically.

Find Points Of Intersection of Ellipse and Line. An interactive calculator to calculate the coordinates of the points of intersection of an ellipse and a line.