Ellipse Standard Form Calculator

An ellipse is defined by its center point (h, k), horizontal semi-axis 'a', and vertical semi-axis 'b'. The standard equation for a horizontally oriented ellipse is given by ((x - h)^2 / a^2) + ((y - k)^2 / b^2) = 1. If the vertical semi-axis is longer, 'a' and 'b' switch positions. The vertices lie at (h ± a, k) for horizontal alignment, while co-vertices (semivertices) sit at (h, k ± b). Foci locations are determined using the fundamental ellipse relationship c^2 = |a^2 - b^2|, where 'c' represents the distance from the center to each focus along the major axis.

Consider an ellipse centered at the origin (0, 0) with a horizontal semi-axis a = 5 and a vertical semi-axis b = 3. First, we plug these values into the standard equation template: (x^2 / 5^2) + (y^2 / 3^2) = 1, which simplifies to (x^2 / 25) + (y^2 / 9) = 1. Next, to find the vertices along the x-axis, we calculate (0 ± 5, 0), yielding Vertex 1 at (-5, 0) and Vertex 2 at (5, 0). For the semivertices along the y-axis, we calculate (0, 0 ± 3), resulting in (0, -3) and (0, 3). Finally, to locate the foci, we compute c = sqrt(5^2 - 3^2) = sqrt(25 - 9) = sqrt(16) = 4, placing our foci at (-4, 0) and (4, 0).

Always double-check whether your ellipse is horizontally or vertically oriented before identifying its major and minor axes; the larger denominator in the standard equation always dictates the major axis. When converting from general quadratic form to standard form, make sure to complete the square carefully for both the x and y variable groupings simultaneously.