Properties of the ellipse $$$2 x^{2} + 9 y^{2} = 18$$$
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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 $$$2 x^{2} + 9 y^{2} = 18$$$.
Solution
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 $$$\frac{\left(x - 0\right)^{2}}{9} + \frac{\left(y - 0\right)^{2}}{2} = 1$$$.
Thus, $$$h = 0$$$, $$$k = 0$$$, $$$a = 3$$$, $$$b = \sqrt{2}$$$.
The standard form is $$$\frac{x^{2}}{3^{2}} + \frac{y^{2}}{\left(\sqrt{2}\right)^{2}} = 1$$$.
The vertex form is $$$\frac{x^{2}}{9} + \frac{y^{2}}{2} = 1$$$.
The general form is $$$2 x^{2} + 9 y^{2} - 18 = 0$$$.
The linear eccentricity (focal distance) is $$$c = \sqrt{a^{2} - b^{2}} = \sqrt{7}$$$.
The eccentricity is $$$e = \frac{c}{a} = \frac{\sqrt{7}}{3}$$$.
The first focus is $$$\left(h - c, k\right) = \left(- \sqrt{7}, 0\right)$$$.
The second focus is $$$\left(h + c, k\right) = \left(\sqrt{7}, 0\right)$$$.
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 $$$\left(h, k - b\right) = \left(0, - \sqrt{2}\right)$$$.
The second co-vertex is $$$\left(h, k + b\right) = \left(0, \sqrt{2}\right)$$$.
The length of the major axis is $$$2 a = 6$$$.
The length of the minor axis is $$$2 b = 2 \sqrt{2}$$$.
The area is $$$\pi a b = 3 \sqrt{2} \pi$$$.
The circumference is $$$4 a E\left(\frac{\pi}{2}\middle| e^{2}\right) = 12 E\left(\frac{7}{9}\right)$$$.
The focal parameter is the distance between the focus and the directrix: $$$\frac{b^{2}}{c} = \frac{2 \sqrt{7}}{7}$$$.
The latera recta are the lines parallel to the minor axis that pass through the foci.
The first latus rectum is $$$x = - \sqrt{7}$$$.
The second latus rectum is $$$x = \sqrt{7}$$$.
The endpoints of the first latus rectum can be found by solving the system $$$\begin{cases} 2 x^{2} + 9 y^{2} - 18 = 0 \\ x = - \sqrt{7} \end{cases}$$$ (for steps, see system of equations calculator).
The endpoints of the first latus rectum are $$$\left(- \sqrt{7}, - \frac{2}{3}\right)$$$, $$$\left(- \sqrt{7}, \frac{2}{3}\right)$$$.
The endpoints of the second latus rectum can be found by solving the system $$$\begin{cases} 2 x^{2} + 9 y^{2} - 18 = 0 \\ x = \sqrt{7} \end{cases}$$$ (for steps, see system of equations calculator).
The endpoints of the second latus rectum are $$$\left(\sqrt{7}, - \frac{2}{3}\right)$$$, $$$\left(\sqrt{7}, \frac{2}{3}\right)$$$.
The length of the latera recta (focal width) is $$$\frac{2 b^{2}}{a} = \frac{4}{3}$$$.
The first directrix is $$$x = h - \frac{a^{2}}{c} = - \frac{9 \sqrt{7}}{7}$$$.
The second directrix is $$$x = h + \frac{a^{2}}{c} = \frac{9 \sqrt{7}}{7}$$$.
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: $$$\left(0, - \sqrt{2}\right)$$$, $$$\left(0, \sqrt{2}\right)$$$
The domain is $$$\left[h - a, h + a\right] = \left[-3, 3\right]$$$.
The range is $$$\left[k - b, k + b\right] = \left[- \sqrt{2}, \sqrt{2}\right]$$$.
Answer
Standard form/equation: $$$\frac{x^{2}}{3^{2}} + \frac{y^{2}}{\left(\sqrt{2}\right)^{2}} = 1$$$A.
Vertex form/equation: $$$\frac{x^{2}}{9} + \frac{y^{2}}{2} = 1$$$A.
General form/equation: $$$2 x^{2} + 9 y^{2} - 18 = 0$$$A.
First focus-directrix form/equation: $$$\left(x + \sqrt{7}\right)^{2} + y^{2} = \frac{7 \left(x + \frac{9 \sqrt{7}}{7}\right)^{2}}{9}$$$A.
Second focus-directrix form/equation: $$$\left(x - \sqrt{7}\right)^{2} + y^{2} = \frac{7 \left(x - \frac{9 \sqrt{7}}{7}\right)^{2}}{9}$$$A.
Graph: see the graphing calculator.
Center: $$$\left(0, 0\right)$$$A.
First focus: $$$\left(- \sqrt{7}, 0\right)\approx \left(-2.645751311064591, 0\right)$$$A.
Second focus: $$$\left(\sqrt{7}, 0\right)\approx \left(2.645751311064591, 0\right)$$$A.
First vertex: $$$\left(-3, 0\right)$$$A.
Second vertex: $$$\left(3, 0\right)$$$A.
First co-vertex: $$$\left(0, - \sqrt{2}\right)\approx \left(0, -1.414213562373095\right)$$$A.
Second co-vertex: $$$\left(0, \sqrt{2}\right)\approx \left(0, 1.414213562373095\right)$$$A.
Major axis length: $$$6$$$A.
Semi-major axis length: $$$3$$$A.
Minor axis length: $$$2 \sqrt{2}\approx 2.82842712474619$$$A.
Semi-minor axis length: $$$\sqrt{2}\approx 1.414213562373095$$$A.
Area: $$$3 \sqrt{2} \pi\approx 13.328648814475099$$$A.
Circumference: $$$12 E\left(\frac{7}{9}\right)\approx 14.318823491478567$$$A.
First latus rectum: $$$x = - \sqrt{7}\approx -2.645751311064591$$$A.
Second latus rectum: $$$x = \sqrt{7}\approx 2.645751311064591$$$A.
Endpoints of the first latus rectum: $$$\left(- \sqrt{7}, - \frac{2}{3}\right)\approx \left(-2.645751311064591, -0.666666666666667\right)$$$, $$$\left(- \sqrt{7}, \frac{2}{3}\right)\approx \left(-2.645751311064591, 0.666666666666667\right)$$$A.
Endpoints of the second latus rectum: $$$\left(\sqrt{7}, - \frac{2}{3}\right)\approx \left(2.645751311064591, -0.666666666666667\right)$$$, $$$\left(\sqrt{7}, \frac{2}{3}\right)\approx \left(2.645751311064591, 0.666666666666667\right)$$$A.
Length of the latera recta (focal width): $$$\frac{4}{3}\approx 1.333333333333333$$$A.
Focal parameter: $$$\frac{2 \sqrt{7}}{7}\approx 0.755928946018454$$$A.
Eccentricity: $$$\frac{\sqrt{7}}{3}\approx 0.881917103688197$$$A.
Linear eccentricity (focal distance): $$$\sqrt{7}\approx 2.645751311064591$$$A.
First directrix: $$$x = - \frac{9 \sqrt{7}}{7}\approx -3.401680257083045$$$A.
Second directrix: $$$x = \frac{9 \sqrt{7}}{7}\approx 3.401680257083045$$$A.
x-intercepts: $$$\left(-3, 0\right)$$$, $$$\left(3, 0\right)$$$A.
y-intercepts: $$$\left(0, - \sqrt{2}\right)\approx \left(0, -1.414213562373095\right)$$$, $$$\left(0, \sqrt{2}\right)\approx \left(0, 1.414213562373095\right)$$$A.
Domain: $$$\left[-3, 3\right]$$$A.
Range: $$$\left[- \sqrt{2}, \sqrt{2}\right]\approx \left[-1.414213562373095, 1.414213562373095\right]$$$A.