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

The solution to 18x^2+24y+12y^2-12x=1 is Ellipse with (h,k)=(1/3 ,-1),a=sqrt(5/6),b=(sqrt(5))/2

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

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

(h, k) = (\frac{1}{3}, - 1), a = \frac{5}{6}, b = \frac{5}{2}

Solution steps

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

Rewrite

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

in the form of the standard ellipse equation

\frac{( x - \frac{1}{3} ) ^{2}}{( \frac{5}{6} ) ^{2}} + \frac{( y - ( - 1 ) ) ^{2}}{( \frac{5}{2} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (\frac{1}{3}, - 1), a = \frac{5}{6}, b = \frac{5}{2}

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (\frac{1}{3}, - 1), b = \frac{5}{2}, a = \frac{5}{6}

What is 18x^2+24y+12y^2-12x=1 ?
The solution to 18x^2+24y+12y^2-12x=1 is Ellipse with (h,k)=(1/3 ,-1),a=sqrt(5/6),b=(sqrt(5))/2