Properties of the ellipse $$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$

The calculator will find the properties of the ellipse $$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$, 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 $$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$.

Solution

This is the slanted ellipse: rotate it by $$$45^0$$$ clockwise.

The new ellipse is $$$6 x^{2} - 30 \sqrt{2} x + 8 y^{2} - 8 \sqrt{2} y + 67 = 0$$$, find its properties (for steps, see ellipse calculator).

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

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

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

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

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

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

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

Major axis length: $$$2 \sqrt{2}$$$.

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

Minor axis length: $$$\sqrt{6}$$$.

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

Area: $$$\sqrt{3} \pi$$$.

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

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

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

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

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

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

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

Eccentricity: $$$\frac{1}{2}$$$.

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

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

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

Now, rotate back.

$$$\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$ becomes $$$\left(2, 3\right)$$$ (for steps, see rotation calculator).

$$$\left(2 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$$$ becomes $$$\left(\frac{3}{2}, \frac{5}{2}\right)$$$ (for steps, see rotation calculator).

$$$\left(3 \sqrt{2}, \frac{\sqrt{2}}{2}\right)$$$ becomes $$$\left(\frac{5}{2}, \frac{7}{2}\right)$$$ (for steps, see rotation calculator).

$$$\left(\frac{3 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$ becomes $$$\left(1, 2\right)$$$ (for steps, see rotation calculator).

$$$\left(\frac{7 \sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$$$ becomes $$$\left(3, 4\right)$$$ (for steps, see rotation calculator).

$$$\left(\frac{5 \sqrt{2}}{2}, \frac{- \sqrt{6} + \sqrt{2}}{2}\right)$$$ becomes $$$\left(\frac{\sqrt{3} + 4}{2}, 3 - \frac{\sqrt{3}}{2}\right)$$$ (for steps, see rotation calculator).

$$$\left(\frac{5 \sqrt{2}}{2}, \frac{\sqrt{2} + \sqrt{6}}{2}\right)$$$ becomes $$$\left(2 - \frac{\sqrt{3}}{2}, \frac{\sqrt{3} + 6}{2}\right)$$$ (for steps, see rotation calculator).

$$$x = 2 \sqrt{2}$$$ becomes $$$\frac{\sqrt{2} \left(x + y\right)}{2} = 2 \sqrt{2}$$$ or $$$y = 4 - x$$$.

$$$x = 3 \sqrt{2}$$$ becomes $$$\frac{\sqrt{2} \left(x + y\right)}{2} = 3 \sqrt{2}$$$ or $$$y = 6 - x$$$.

$$$\left(2 \sqrt{2}, - \frac{\sqrt{2}}{4}\right)$$$ becomes $$$\left(\frac{9}{4}, \frac{7}{4}\right)$$$ (for steps, see rotation calculator).

$$$\left(2 \sqrt{2}, \frac{5 \sqrt{2}}{4}\right)$$$ becomes $$$\left(\frac{3}{4}, \frac{13}{4}\right)$$$ (for steps, see rotation calculator).

$$$\left(3 \sqrt{2}, - \frac{\sqrt{2}}{4}\right)$$$ becomes $$$\left(\frac{13}{4}, \frac{11}{4}\right)$$$ (for steps, see rotation calculator).

$$$\left(3 \sqrt{2}, \frac{5 \sqrt{2}}{4}\right)$$$ becomes $$$\left(\frac{7}{4}, \frac{17}{4}\right)$$$ (for steps, see rotation calculator).

$$$x = \frac{\sqrt{2}}{2}$$$ becomes $$$\frac{\sqrt{2} \left(x + y\right)}{2} = \frac{\sqrt{2}}{2}$$$ or $$$y = 1 - x$$$.

$$$x = \frac{9 \sqrt{2}}{2}$$$ becomes $$$\frac{\sqrt{2} \left(x + y\right)}{2} = \frac{9 \sqrt{2}}{2}$$$ or $$$y = 9 - x$$$.

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

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

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.

Answer

General form/equation: $$$7 x^{2} - 2 x y - 22 x + 7 y^{2} - 38 y + 67 = 0$$$A.

First focus-directrix form/equation: $$$\left(x - \frac{3}{2}\right)^{2} + \left(y - \frac{5}{2}\right)^{2} = \frac{\left(x + y - 1\right)^{2}}{8}$$$A.

Second focus-directrix form/equation: $$$\left(x - \frac{5}{2}\right)^{2} + \left(y - \frac{7}{2}\right)^{2} = \frac{\left(x + y - 9\right)^{2}}{8}$$$A.

Graph: see the graphing calculator.

Center: $$$\left(2, 3\right)$$$A.

First focus: $$$\left(\frac{3}{2}, \frac{5}{2}\right) = \left(1.5, 2.5\right)$$$A.

Second focus: $$$\left(\frac{5}{2}, \frac{7}{2}\right) = \left(2.5, 3.5\right)$$$A.

First vertex: $$$\left(1, 2\right)$$$A.

Second vertex: $$$\left(3, 4\right)$$$A.

First co-vertex: $$$\left(\frac{\sqrt{3} + 4}{2}, 3 - \frac{\sqrt{3}}{2}\right)\approx \left(2.866025403784439, 2.133974596215561\right).$$$A

Second co-vertex: $$$\left(2 - \frac{\sqrt{3}}{2}, \frac{\sqrt{3} + 6}{2}\right)\approx \left(1.133974596215561, 3.866025403784439\right).$$$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: $$$y = 4 - x$$$A.

Second latus rectum: $$$y = 6 - x$$$A.

Endpoints of the first latus rectum: $$$\left(\frac{9}{4}, \frac{7}{4}\right) = \left(2.25, 1.75\right)$$$, $$$\left(\frac{3}{4}, \frac{13}{4}\right) = \left(0.75, 3.25\right)$$$A.

Endpoints of the second latus rectum: $$$\left(\frac{13}{4}, \frac{11}{4}\right) = \left(3.25, 2.75\right)$$$, $$$\left(\frac{7}{4}, \frac{17}{4}\right) = \left(1.75, 4.25\right)$$$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: $$$y = 1 - x$$$A.

Second directrix: $$$y = 9 - x$$$A.

x-intercepts: no x-intercepts.

y-intercepts: no y-intercepts.

Domain: $$$\left[2 - \frac{\sqrt{7}}{2}, \frac{\sqrt{7} + 4}{2}\right]\approx \left[0.677124344467705, 3.322875655532295\right].$$$A

Range: $$$\left[3 - \frac{\sqrt{7}}{2}, \frac{\sqrt{7} + 6}{2}\right]\approx \left[1.677124344467705, 4.322875655532295\right].$$$A