What is the eccentricity 18y^2+24y+12x^2-12x=1 ?

The eccentricity 18y^2+24y+12x^2-12x=1 is 1/(sqrt(3))

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eccentricity 18y^2+24y+12x^2-12x=1

eccentricity

18 y^{2} + 24 y + 12 x^{2} - 12 x = 1

\frac{1}{3}

Solution steps

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

Calculate ellipse properties

18 y^{2} + 24 y + 12 x^{2} - 12 x = 1 :

Ellipse with center

(h, k) = (\frac{1}{2}, - \frac{2}{3}), a = 1, b = \frac{2}{3}

= \frac{1 ^{2} - ( \frac{2}{3} ) ^{2}}{1}

\frac{1 ^{2} - ( \frac{2}{3} ) ^{2}}{1} = \frac{1}{3}

= \frac{1}{3}

+ 9 x^{2} + 4 y^{2} = 36

9 x^{2} + 4 y^{2} + 54 x - 8 y + 49 = 0

(\frac{x}{8})^{2} + (\frac{y}{6})^{2} = 1

4 x^{2} + 9 y^{2} - 8 x - 32 = 0

9 x^{2} + 6 y^{2} = 54

What is the eccentricity 18y^2+24y+12x^2-12x=1 ?
The eccentricity 18y^2+24y+12x^2-12x=1 is 1/(sqrt(3))