Ellipse Calculator

Find the center, foci, vertices, co-vertices, major axis length, semi-major axis length, minor axis length, semi-minor axis length, area, circumference, latera recta, length of the latera recta (focal width), focal parameter, eccentricity, linear eccentricity (focal distance), directrices, x-intercepts, y-intercepts, domain, and range of the ellipse $4 x^{2} + 9 y^{2} = 36$.

The equation of an ellipse is $\frac{\left(x - h\right)^{2}}{a^{2}} + \frac{\left(y - k\right)^{2}}{b^{2}} = 1$, where $\left(h, k\right)$ is the center, $a$ and $b$ are the lengths of the semi-major and the semi-minor axes.

Our ellipse in this form is $\frac{\left(x - 0\right)^{2}}{9} + \frac{\left(y - 0\right)^{2}}{4} = 1$.

Thus, $h = 0$, $k = 0$, $a = 3$, $b = 2$.

The standard form is $\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1$.

The vertex form is $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$.

The general form is $4 x^{2} + 9 y^{2} - 36 = 0$.

The linear eccentricity (focal distance) is $c = \sqrt{a^{2} - b^{2}} = \sqrt{5}$.

The eccentricity is $e = \frac{c}{a} = \frac{\sqrt{5}}{3}$.

The first focus is $\left(h - c, k\right) = \left(- \sqrt{5}, 0\right)$.

The second focus is $\left(h + c, k\right) = \left(\sqrt{5}, 0\right)$.

The first vertex is $\left(h - a, k\right) = \left(-3, 0\right)$.

The second vertex is $\left(h + a, k\right) = \left(3, 0\right)$.

The first co-vertex is $\left(h, k - b\right) = \left(0, -2\right)$.

The second co-vertex is $\left(h, k + b\right) = \left(0, 2\right)$.

The length of the major axis is $2 a = 6$.

The length of the minor axis is $2 b = 4$.

The area is $\pi a b = 6 \pi$.

The circumference is $4 a E\left(\frac{\pi}{2}\middle| e^{2}\right) = 12 E\left(\frac{5}{9}\right)$.

The focal parameter is the distance between the focus and the directrix: $\frac{b^{2}}{c} = \frac{4 \sqrt{5}}{5}$.

The latera recta are the lines parallel to the minor axis that pass through the foci.

The first latus rectum is $x = - \sqrt{5}$.

The second latus rectum is $x = \sqrt{5}$.

The endpoints of the first latus rectum can be found by solving the system $\begin{cases} 4 x^{2} + 9 y^{2} - 36 = 0 \\ x = - \sqrt{5} \end{cases}$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $\left(- \sqrt{5}, - \frac{4}{3}\right)$, $\left(- \sqrt{5}, \frac{4}{3}\right)$.

The endpoints of the second latus rectum can be found by solving the system $\begin{cases} 4 x^{2} + 9 y^{2} - 36 = 0 \\ x = \sqrt{5} \end{cases}$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $\left(\sqrt{5}, - \frac{4}{3}\right)$, $\left(\sqrt{5}, \frac{4}{3}\right)$.

The length of the latera recta (focal width) is $\frac{2 b^{2}}{a} = \frac{8}{3}$.

The first directrix is $x = h - \frac{a^{2}}{c} = - \frac{9 \sqrt{5}}{5}$.

The second directrix is $x = h + \frac{a^{2}}{c} = \frac{9 \sqrt{5}}{5}$.

The x-intercepts can be found by setting $y = 0$ in the equation and solving for $x$ (for steps, see intercepts calculator).

x-intercepts: $\left(-3, 0\right)$, $\left(3, 0\right)$

The y-intercepts can be found by setting $x = 0$ in the equation and solving for $y$: (for steps, see intercepts calculator).

y-intercepts: $\left(0, -2\right)$, $\left(0, 2\right)$

The domain is $\left[h - a, h + a\right] = \left[-3, 3\right]$.

The range is $\left[k - b, k + b\right] = \left[-2, 2\right]$.

Standard form/equation: $\frac{x^{2}}{3^{2}} + \frac{y^{2}}{2^{2}} = 1$A.

Vertex form/equation: $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$A.

General form/equation: $4 x^{2} + 9 y^{2} - 36 = 0$A.

First focus-directrix form/equation: $\left(x + \sqrt{5}\right)^{2} + y^{2} = \frac{5 \left(x + \frac{9 \sqrt{5}}{5}\right)^{2}}{9}$A.

Second focus-directrix form/equation: $\left(x - \sqrt{5}\right)^{2} + y^{2} = \frac{5 \left(x - \frac{9 \sqrt{5}}{5}\right)^{2}}{9}$A.

Graph: see the graphing calculator.

Center: $\left(0, 0\right)$A.

First focus: $\left(- \sqrt{5}, 0\right)\approx \left(-2.23606797749979, 0\right)$A.

Second focus: $\left(\sqrt{5}, 0\right)\approx \left(2.23606797749979, 0\right)$A.

First vertex: $\left(-3, 0\right)$A.

Second vertex: $\left(3, 0\right)$A.

First co-vertex: $\left(0, -2\right)$A.

Second co-vertex: $\left(0, 2\right)$A.

Major axis length: $6$A.

Semi-major axis length: $3$A.

Minor axis length: $4$A.

Semi-minor axis length: $2$A.

Area: $6 \pi\approx 18.849555921538759$A.

Circumference: $12 E\left(\frac{5}{9}\right)\approx 15.86543958929059$A.

First latus rectum: $x = - \sqrt{5}\approx -2.23606797749979$A.

Second latus rectum: $x = \sqrt{5}\approx 2.23606797749979$A.

Endpoints of the first latus rectum: $\left(- \sqrt{5}, - \frac{4}{3}\right)\approx \left(-2.23606797749979, -1.333333333333333\right)$, $\left(- \sqrt{5}, \frac{4}{3}\right)\approx \left(-2.23606797749979, 1.333333333333333\right)$A.

Endpoints of the second latus rectum: $\left(\sqrt{5}, - \frac{4}{3}\right)\approx \left(2.23606797749979, -1.333333333333333\right)$, $\left(\sqrt{5}, \frac{4}{3}\right)\approx \left(2.23606797749979, 1.333333333333333\right)$A.

Length of the latera recta (focal width): $\frac{8}{3}\approx 2.666666666666667$A.

Focal parameter: $\frac{4 \sqrt{5}}{5}\approx 1.788854381999832$A.

Eccentricity: $\frac{\sqrt{5}}{3}\approx 0.74535599249993$A.

Linear eccentricity (focal distance): $\sqrt{5}\approx 2.23606797749979$A.

First directrix: $x = - \frac{9 \sqrt{5}}{5}\approx -4.024922359499621$A.

Second directrix: $x = \frac{9 \sqrt{5}}{5}\approx 4.024922359499621$A.

x-intercepts: $\left(-3, 0\right)$, $\left(3, 0\right)$A.

y-intercepts: $\left(0, -2\right)$, $\left(0, 2\right)$A.

Domain: $\left[-3, 3\right]$A.

Range: $\left[-2, 2\right]$A.