Ellipse Equation Calculator
Find properties of an ellipse from its equation.
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.
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$
How to use this calculator
- Enter the ellipse equation in standard form.
- Identify the center coordinates from the shifted x and y terms.
- Compare the denominators to find the semi-axis lengths and the major axis direction.
- Use the calculator output to read the ellipse properties such as center, vertices, and foci.
The formula explained
$$ \frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1 $$
- \(x\) = 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
- Rewrite the equation so the left side matches the standard ellipse form with a 1 on the right.
- Read the center directly from the shifts in x and y.
- Take the square roots of the denominators to get the semi-axis lengths.
- Use the larger denominator to decide whether the major axis is horizontal or vertical.
- Use the calculated values to identify the vertices, co-vertices, and foci.
Worked example
Suppose you want the properties of the ellipse given by \(\frac{(x-3)^2}{16}+\frac{(y+2)^2}{9}=1\).
- Match the equation to standard form. The center is \((3,-2)\) because the x term is \(x-3\) and the y term is \(y+2\).
- 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.
- Compute the focal distance using \(c^2=a^2-b^2=16-9=7\), so \(c=\sqrt{7}\).
Answer. Center: (3, -2), major axis horizontal, semi-major axis 4, semi-minor axis 3, foci: (3 - sqrt(7), -2) and (3 + sqrt(7), -2).
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
What kind of equation does this calculator need?+
It works with an ellipse written in standard form. The equation should show the squared x and y terms divided by positive numbers and set equal to 1. If the equation is not in that form yet, rewrite it first.
What properties can I learn from the result?+
You can usually find the center, the lengths of the semi-axes, the direction of the major axis, and the locations of the vertices and foci. These are the main features that describe the ellipse.
How do I know if the major axis is horizontal or vertical?+
Look at which squared term has the larger denominator. If the x term has the larger denominator, the major axis is horizontal. If the y term has the larger denominator, the major axis is vertical.
Why is my answer not showing correctly?+
The most common issue is entering an equation that is not in standard form. Another common problem is using a negative denominator or forgetting to make the right side equal to 1. Check the format carefully before trying again.
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