Ellipse Equation Calculator

About Ellipse Equation Calculator

An ellipse has two foci. The sum of distances from any point on the ellipse to both foci equals 2a. Eccentricity 0 < e < 1.

How to use this calculator

1. Enter the ellipse equation in standard form.
2. Identify the center coordinates from the shifted x and y terms.
3. Compare the denominators to find the semi-axis lengths and the major axis direction.
4. Use the calculator output to read the ellipse properties such as center, vertices, and foci.

The formula explained

• \(x\) = the x-coordinate of a point on the ellipse
• \(y\) = the y-coordinate of a point on the ellipse
• \(h\) = the x-coordinate of the center
• \(k\) = the y-coordinate of the center
• \(a\) = the semi-axis length along the horizontal direction when the major axis is horizontal, or the larger semi-axis length in general form
• \(b\) = the semi-axis length along the vertical direction when the major axis is horizontal, or the smaller semi-axis length in general form

Step by step method

1. Rewrite the equation so the left side matches the standard ellipse form with a 1 on the right.
2. Read the center directly from the shifts in x and y.
3. Take the square roots of the denominators to get the semi-axis lengths.
4. Use the larger denominator to decide whether the major axis is horizontal or vertical.
5. Use the calculated values to identify the vertices, co-vertices, and foci.

Worked example

Suppose you want the properties of the ellipse given by \(\frac{(x-3)^2}{16}+\frac{(y+2)^2}{9}=1\).

1. Match the equation to standard form. The center is \((3,-2)\) because the x term is \(x-3\) and the y term is \(y+2\).
2. Find the semi-axis lengths: \(a=\sqrt{16}=4\) and \(b=\sqrt{9}=3\). Since 16 is larger than 9, the major axis is horizontal.
3. Compute the focal distance using \(c^2=a^2-b^2=16-9=7\), so \(c=\sqrt{7}\).

Tips and common mistakes

• The larger denominator tells you the direction of the major axis.
• Do not confuse the center shifts with the signs in the equation. x - h means the center x-coordinate is h, while y - k means the center y-coordinate is k.
• Remember that the semi-axis lengths are the square roots of the denominators, not the denominators themselves.
• If the equation is not already set equal to 1, simplify or divide first before using the calculator.

Frequently asked questions