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.

Type:

Equation:

Center:$$$($$$

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First focus:$$$($$$

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Second focus:$$$($$$

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First vertex:$$$($$$

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Second vertex:$$$($$$

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First co-vertex:$$$($$$

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Second co-vertex:$$$($$$

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Length of the major axis:

If you have the length of the semi-major axis (a), enter its value multiplied by $$$2$$$ here.

Length of the minor axis:

If you have the length of the semi-minor axis (b), enter its value multiplied by $$$2$$$ here.

Eccentricity:

Area:

First directrix:

In any form you want: $$$y = -6$$$, $$$y = 2 x + 5$$$, etc.

Second directrix:

In any form you want: $$$x - 3 = 0$$$, $$$- 8 x + 7 y + 1 = 0$$$, etc.

First point:$$$($$$

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Second point:$$$($$$

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Third point:$$$($$$

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Fourth point:$$$($$$

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Vertical major axis (parallel to the y-axis)?

Horizontal minor axis (parallel to the x-axis)?

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Your Input

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

Answer

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

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

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