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

The solution to 18x^2+24x+12y^2-12y=1 is Ellipse with (h,k)=(-2/3 , 1/2),a=sqrt(2/3),b=1

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

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

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

Solution steps

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

Rewrite

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

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

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

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

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

16 x^{2} + 25 y^{2} - 64 x - 50 y - 311 = 0

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

2 x^{2} + y^{2} - 7 = 0

x^{2} + 2 y^{2} - 8 x + 8 y = 20

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

What is 18x^2+24x+12y^2-12y=1 ?
The solution to 18x^2+24x+12y^2-12y=1 is Ellipse with (h,k)=(-2/3 , 1/2),a=sqrt(2/3),b=1