What is 12x^2+3y^2+12y-96=0 ?

The solution to 12x^2+3y^2+12y-96=0 is Ellipse with (h,k)=(0,-2),a=3,b=6

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12x^2+3y^2+12y-96=0

12 x^{2} + 3 y^{2} + 12 y - 96 = 0

(h, k) = (0, - 2), a = 3, b = 6

Solution steps

12 x^{2} + 3 y^{2} + 12 y - 96 = 0

Rewrite

12 x^{2} + 3 y^{2} + 12 y - 96 = 0

in the form of the standard ellipse equation

\frac{( x - 0 ) ^{2}}{3 ^{2}} + \frac{( y - ( - 2 ) ) ^{2}}{6 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (0, - 2), a = 3, b = 6

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (0, - 2), b = 6, a = 3

\frac{( x - 3 ) ^{2}}{2} + (y + 5)^{2} = 8

(\frac{x - 2}{9})^{2} + (\frac{y + 2}{25})^{2} = 1

25 x^{2} + 64 y^{2} - 400 x - 640 y + 1600 = 0

9 x^{2} + 25 y^{2} = 64

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

What is 12x^2+3y^2+12y-96=0 ?
The solution to 12x^2+3y^2+12y-96=0 is Ellipse with (h,k)=(0,-2),a=3,b=6