What is the eccentricity 3x^2+y^2+2x-6y=3 ?

The eccentricity 3x^2+y^2+2x-6y=3 is sqrt(2/3)

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eccentricity 3x^2+y^2+2x-6y=3

eccentricity

3 x^{2} + y^{2} + 2 x - 6 y = 3

\frac{2}{3}

Solution steps

= \frac{b ^{2} - a ^{2}}{b}

Calculate ellipse properties

3 x^{2} + y^{2} + 2 x - 6 y = 3 :

Ellipse with center

(h, k) = (- \frac{1}{3}, 3), b = \frac{37}{3}, a = \frac{37}{3}

= \frac{( \frac{37}{3} ) ^{2} - ( \frac{37}{3} ) ^{2}}{\frac{37}{3}}

\frac{( \frac{37}{3} ) ^{2} - ( \frac{37}{3} ) ^{2}}{\frac{37}{3}} = \frac{2}{3}

= \frac{2}{3}

3 x^{2} + 12 x + 4 y^{2} - 40 y + 100 = 0

4 x^{2} + 9 y^{2} + 32 x - 18 y + 37 = 0

25 x^{2} + 9 y^{2} = 225

5 x^{2} + 7 y^{2} = 70

5 x^{2} + 10 y^{2} = 50

What is the eccentricity 3x^2+y^2+2x-6y=3 ?
The eccentricity 3x^2+y^2+2x-6y=3 is sqrt(2/3)