Enter the center and semi-axes — get the standard equation, foci, vertices, eccentricity and area of the ellipse, drawn to scale.
Enter the center and the two semi-axes, and the analyzer writes the standard equation, places the foci and vertices, and graphs the ellipse.
An ellipse centered at (h, k) with horizontal semi-axis a and vertical semi-axis b is (x − h)²/a² + (y − k)²/b² = 1. The larger of a and b is the semi-major axis and sets the orientation: wide when a > b, tall when b > a.
With the semi-major axis called A and the semi-minor called B, the focal distance is c = √(A² − B²). The two foci sit c units from the center along the major axis, and the vertices sit A units out along the same axis.
Eccentricity e = c / A measures how stretched the ellipse is: 0 is a perfect circle and values close to 1 are long and thin. The area is πab. The analyzer shows each of these as its own step and draws the foci and vertices on the graph.