What is the eccentricity 4x^2+3y^2+8x-24y+51=0 ?

The eccentricity 4x^2+3y^2+8x-24y+51=0 is 1/2

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eccentricity 4x^2+3y^2+8x-24y+51=0

eccentricity

4 x^{2} + 3 y^{2} + 8 x - 24 y + 51 = 0

\frac{1}{2}

Solution steps

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

Calculate ellipse properties

4 x^{2} + 3 y^{2} + 8 x - 24 y + 51 = 0 :

Ellipse with center

(h, k) = (- 1, 4), b = \frac{1}{3}, a = \frac{1}{2}

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

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

= \frac{1}{2}

x^{2} + y^{2} + 6 x - 4 y - 3 = 0

\frac{x ^{2}}{25} + \frac{y ^{2}}{36} = 1

4 x^{2} + y^{2} = 16

x^{2} + y^{2} - 6 x - 8 y = 50

\frac{x ^{2}}{3 ^{2}} + \frac{y ^{2}}{9 ^{2}} = 1

What is the eccentricity 4x^2+3y^2+8x-24y+51=0 ?
The eccentricity 4x^2+3y^2+8x-24y+51=0 is 1/2