# Properties of the ellipse $6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0$

The calculator will find the properties of the ellipse $6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0$, with steps shown.

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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 $6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0$.

### 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 $\frac{\left(x - \frac{5 \sqrt{2}}{2}\right)^{2}}{2} + \frac{\left(y - \frac{\sqrt{2}}{2}\right)^{2}}{\frac{3}{2}} = 1$.

Thus, $h = \frac{5 \sqrt{2}}{2}$, $k = \frac{\sqrt{2}}{2}$, $a = \sqrt{2}$, $b = \frac{\sqrt{6}}{2}$.

The standard form is $\frac{\left(x - \frac{5 \sqrt{2}}{2}\right)^{2}}{\left(\sqrt{2}\right)^{2}} + \frac{\left(y - \frac{\sqrt{2}}{2}\right)^{2}}{\left(\frac{\sqrt{6}}{2}\right)^{2}} = 1$.

The vertex form is $\frac{\left(x - \frac{5 \sqrt{2}}{2}\right)^{2}}{2} + \frac{2 \left(y - \frac{\sqrt{2}}{2}\right)^{2}}{3} = 1$.

The general form is $6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0$.

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

The eccentricity is $e = \frac{c}{a} = \frac{1}{2}$.

The first focus is $\left(h - c, k\right) = \left(2 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$.

The second focus is $\left(h + c, k\right) = \left(3 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$.

The first vertex is $\left(h - a, k\right) = \left(\frac{3 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

The second vertex is $\left(h + a, k\right) = \left(\frac{7 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

The first co-vertex is $\left(h, k - b\right) = \left(\frac{5 \sqrt{2}}{2}, \frac{- \sqrt{6} + \sqrt{2}}{2}\right)$.

The second co-vertex is $\left(h, k + b\right) = \left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2} + \sqrt{6}}{2}\right)$.

The length of the major axis is $2 a = 2 \sqrt{2}$.

The length of the minor axis is $2 b = \sqrt{6}$.

The area is $\pi a b = \sqrt{3} \pi$.

The circumference is $4 a E\left(\frac{\pi}{2}\middle| e^{2}\right) = 4 \sqrt{2} E\left(\frac{1}{4}\right)$.

The focal parameter is the distance between the focus and the directrix: $\frac{b^{2}}{c} = \frac{3 \sqrt{2}}{2}$.

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

The first latus rectum is $x = 2 \sqrt{2}$.

The second latus rectum is $x = 3 \sqrt{2}$.

The endpoints of the first latus rectum can be found by solving the system $\begin{cases} 6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0 \\ x = 2 \sqrt{2} \end{cases}$ (for steps, see system of equations calculator).

The endpoints of the first latus rectum are $\left(2 \sqrt{2}, - \frac{\sqrt{2}}{4}\right)$, $\left(2 \sqrt{2}, \frac{5 \sqrt{2}}{4}\right)$.

The endpoints of the second latus rectum can be found by solving the system $\begin{cases} 6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0 \\ x = 3 \sqrt{2} \end{cases}$ (for steps, see system of equations calculator).

The endpoints of the second latus rectum are $\left(3 \sqrt{2}, - \frac{\sqrt{2}}{4}\right)$, $\left(3 \sqrt{2}, \frac{5 \sqrt{2}}{4}\right)$.

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

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

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

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

x-intercepts: $\left(- \frac{2 \sqrt{3}}{3} + \frac{5 \sqrt{2}}{2}, 0\right)$, $\left(\frac{2 \sqrt{3}}{3} + \frac{5 \sqrt{2}}{2}, 0\right)$

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

Since there are no real solutions, there are no y-intercepts.

The domain is $\left[h - a, h + a\right] = \left[\frac{3 \sqrt{2}}{2}, \frac{7 \sqrt{2}}{2}\right]$.

The range is $\left[k - b, k + b\right] = \left[\frac{- \sqrt{6} + \sqrt{2}}{2}, \frac{\sqrt{2} + \sqrt{6}}{2}\right]$.

Standard form/equation: $\frac{\left(x - \frac{5 \sqrt{2}}{2}\right)^{2}}{\left(\sqrt{2}\right)^{2}} + \frac{\left(y - \frac{\sqrt{2}}{2}\right)^{2}}{\left(\frac{\sqrt{6}}{2}\right)^{2}} = 1$A.

Vertex form/equation: $\frac{\left(x - \frac{5 \sqrt{2}}{2}\right)^{2}}{2} + \frac{2 \left(y - \frac{\sqrt{2}}{2}\right)^{2}}{3} = 1$A.

General form/equation: $6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0$A.

First focus-directrix form/equation: $\left(x - 2 \sqrt{2}\right)^{2} + \left(y - \frac{\sqrt{2}}{2}\right)^{2} = \frac{\left(x - \frac{\sqrt{2}}{2}\right)^{2}}{4}$A.

Second focus-directrix form/equation: $\left(x - 3 \sqrt{2}\right)^{2} + \left(y - \frac{\sqrt{2}}{2}\right)^{2} = \frac{\left(x - \frac{9 \sqrt{2}}{2}\right)^{2}}{4}$A.

Graph: see the graphing calculator.

Center: $\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\approx \left(3.535533905932738, 0.707106781186548\right)$A.

First focus: $\left(2 \sqrt{2}, \frac{\sqrt{2}}{2}\right)\approx \left(2.82842712474619, 0.707106781186548\right)$A.

Second focus: $\left(3 \sqrt{2}, \frac{\sqrt{2}}{2}\right)\approx \left(4.242640687119285, 0.707106781186548\right)$A.

First vertex: $\left(\frac{3 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\approx \left(2.121320343559643, 0.707106781186548\right)$A.

Second vertex: $\left(\frac{7 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)\approx \left(4.949747468305833, 0.707106781186548\right)$A.

First co-vertex: $\left(\frac{5 \sqrt{2}}{2}, \frac{- \sqrt{6} + \sqrt{2}}{2}\right)\approx \left(3.535533905932738, -0.517638090205042\right).$A

Second co-vertex: $\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2} + \sqrt{6}}{2}\right)\approx \left(3.535533905932738, 1.931851652578137\right).$A

Major axis length: $2 \sqrt{2}\approx 2.82842712474619$A.

Semi-major axis length: $\sqrt{2}\approx 1.414213562373095$A.

Minor axis length: $\sqrt{6}\approx 2.449489742783178$A.

Semi-minor axis length: $\frac{\sqrt{6}}{2}\approx 1.224744871391589$A.

Area: $\sqrt{3} \pi\approx 5.441398092702654$A.

Circumference: $4 \sqrt{2} E\left(\frac{1}{4}\right)\approx 8.301219834871215$A.

First latus rectum: $x = 2 \sqrt{2}\approx 2.82842712474619$A.

Second latus rectum: $x = 3 \sqrt{2}\approx 4.242640687119285$A.

Endpoints of the first latus rectum: $\left(2 \sqrt{2}, - \frac{\sqrt{2}}{4}\right)\approx \left(2.82842712474619, -0.353553390593274\right)$, $\left(2 \sqrt{2}, \frac{5 \sqrt{2}}{4}\right)\approx \left(2.82842712474619, 1.767766952966369\right)$A.

Endpoints of the second latus rectum: $\left(3 \sqrt{2}, - \frac{\sqrt{2}}{4}\right)\approx \left(4.242640687119285, -0.353553390593274\right)$, $\left(3 \sqrt{2}, \frac{5 \sqrt{2}}{4}\right)\approx \left(4.242640687119285, 1.767766952966369\right)$A.

Length of the latera recta (focal width): $\frac{3 \sqrt{2}}{2}\approx 2.121320343559643$A.

Focal parameter: $\frac{3 \sqrt{2}}{2}\approx 2.121320343559643$A.

Eccentricity: $\frac{1}{2} = 0.5$A.

Linear eccentricity (focal distance): $\frac{\sqrt{2}}{2}\approx 0.707106781186548$A.

First directrix: $x = \frac{\sqrt{2}}{2}\approx 0.707106781186548$A.

Second directrix: $x = \frac{9 \sqrt{2}}{2}\approx 6.363961030678928$A.

x-intercepts: $\left(- \frac{2 \sqrt{3}}{3} + \frac{5 \sqrt{2}}{2}, 0\right)\approx \left(2.380833367553486, 0\right)$, $\left(\frac{2 \sqrt{3}}{3} + \frac{5 \sqrt{2}}{2}, 0\right)\approx \left(4.690234444311989, 0\right)$A.

y-intercepts: no y-intercepts.

Domain: $\left[\frac{3 \sqrt{2}}{2}, \frac{7 \sqrt{2}}{2}\right]\approx \left[2.121320343559643, 4.949747468305833\right]$A.

Range: $\left[\frac{- \sqrt{6} + \sqrt{2}}{2}, \frac{\sqrt{2} + \sqrt{6}}{2}\right]\approx \left[-0.517638090205042, 1.931851652578137\right].$A