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

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

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

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

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

Solution steps

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

Rewrite

18 x^{2} + 12 y^{2} + 24 x - 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} + 32 x - 150 y - 159 = 0

(x - 1)^{2} = 4 - 2 (y + 5)^{2}

4 x^{2} - 8 x + 9 y^{2} - 72 y + 112 = 0

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

4 x^{2} + 81 y^{2} = 324

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