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

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

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

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

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

Solution steps

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

Rewrite

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

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

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

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

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

x^{2} + 2 y^{2} - 2 x - 4 y = - 1

2 x^{2} + y^{2} \leq 1

\frac{x ^{2}}{7 ^{2}} + \frac{y ^{2}}{3 ^{2}} = 1

16 x^{2} + 25 y^{2} - 64 x + 150 y = - 279

4 x^{2} + y^{2} - 40 x + 4 y + 68 = 0

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