Ellipse Standard Form Calculator

What Is the Standard Form of an Ellipse?

An ellipse is the set of all points $(x, y)$ in a plane such that the sum of the distances from two fixed points (foci) is constant. In analytic geometry, the standard equation of an ellipse provides immediate insight into its center coordinates, orientation, semi-major axis, semi-minor axis, and focal points.

Standard Form Equations

Depending on whether the major axis is oriented horizontally (parallel to the x-axis) or vertically (parallel to the y-axis), the standard form is written as follows:

1. Horizontal Major Axis

When the major axis is horizontal with center at $(h, k)$:

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (a \ge b > 0)$$

Center: $(h, k)$
Vertices: $(h \pm a, k)$
Co-Vertices: $(h, k \pm b)$
Foci: $(h \pm c, k)$, where $c = \sqrt{a^2 - b^2}$
Directrices: $x = h \pm \frac{a}{e} = h \pm \frac{a^2}{c}$

2. Vertical Major Axis

When the major axis is vertical with center at $(h, k)$:

$$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad (a \ge b > 0)$$

Center: $(h, k)$
Vertices: $(h, k \pm a)$
Co-Vertices: $(h \pm b, k)$
Foci: $(h, k \pm c)$, where $c = \sqrt{a^2 - b^2}$
Directrices: $y = k \pm \frac{a}{e} = k \pm \frac{a^2}{c}$

Converting from General Form to Standard Form

The general second-degree equation of an ellipse aligned with the Cartesian axes is:

$$Ax^2 + Cy^2 + Dx + Ey + F = 0$$

To convert this into standard form, apply the method of completing the square for both the $x$ and $y$ variables:

Group the $x$-terms and $y$-terms: $A(x^2 + \frac{D}{A}x) + C(y^2 + \frac{E}{C}y) = -F$.
Complete the square inside each set of parentheses: $A(x - h)^2 + C(y - k)^2 = \frac{D^2}{4A} + \frac{E^2}{4C} - F = M$.
Divide both sides by $M$ to set the right side equal to 1: $\frac{(x - h)^2}{M/A} + \frac{(y - k)^2}{M/C} = 1$.
Assign $a^2 = \max(M/A, M/C)$ and $b^2 = \min(M/A, M/C)$.

Key Properties & Formulas

Property	Formula
Linear Eccentricity (Focal Distance)	$c = \sqrt{a^2 - b^2}$
Eccentricity	$e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}} \quad (0 \le e < 1)$
Area of Ellipse	$A = \pi a b$
Latus Rectum Length	$2l = \frac{2b^2}{a}$
Perimeter (Ramanujan First Approx)	$P \approx \pi [3(a+b) - \sqrt{(3a+b)(a+3b)}]$

Related Calculators

Explore other geometry and conic section tools:

Ellipse Calculator: Calculate general area, perimeter, and axes.
Center of Ellipse Calculator: Determine center coordinates from equations or endpoints.
Ellipse Circumference Calculator: Exact series and approximation formulas for ellipse perimeter.
Conic Section Identifier: Classify equations into circles, parabolas, ellipses, and hyperbolas.

Frequently Asked Questions

What is the difference between major axis and semi-major axis?

The major axis is the longest diameter running through the center and both foci of the ellipse, spanning a length of $2a$. The semi-major axis $a$ is half of that total distance, measuring from the center to a vertex.

How do you know if an ellipse is horizontal or vertical in standard form?

Look at the denominators under $(x - h)^2$ and $(y - k)^2$. Since $a > b$, the larger denominator represents $a^2$. If the larger denominator is under $(x - h)^2$, the ellipse is horizontal. If the larger denominator is under $(y - k)^2$, the ellipse is vertical.

What happens when a = b in the standard equation?

When $a = b = r$, the focal distance $c = \sqrt{r^2 - r^2} = 0$, and the eccentricity $e = 0$. The equation simplifies to $(x - h)^2 + (y - k)^2 = r^2$, which is the standard equation of a circle of radius $r$.

What is the eccentricity of an ellipse?

Eccentricity ($e = c/a$) measures how elongated or stretched the ellipse is compared to a perfect circle. A circle has $e = 0$, while values approaching $1$ represent increasingly flattened, needle-like ellipses.

How do you find the directrix of an ellipse?

The directrices are two parallel lines perpendicular to the major axis situated at distance $d = a/e = a^2/c$ from the center. For a horizontal ellipse, the equations are $x = h \pm a/e$, and for a vertical ellipse, $y = k \pm a/e$.

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