# 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 $45^{\circ}$ 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.

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