What is y^2+2y+6x^2-12x=1 ?

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

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y^2+2y+6x^2-12x=1

y^{2} + 2 y + 6 x^{2} - 12 x = 1

(h, k) = (1, - 1), a = \frac{2}{3}, b = 22

Solution steps

y^{2} + 2 y + 6 x^{2} - 12 x = 1

Rewrite

y^{2} + 2 y + 6 x^{2} - 12 x = 1

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (1, - 1), a = \frac{2}{3}, b = 22

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (1, - 1), b = 22, a = \frac{2}{3}

x^{2} + 2 y^{2} - 2 x - 4 y - 5 = 0

x^{2} + 9 y^{2} + 90 y + 189 = 0

4 x^{2} + 36 y^{2} = 144

2 x - x^{2} - 4 y^{2} - 16 y - 1 \geq 0

\frac{( x - 3 ) ^{2}}{16} + \frac{( y - 2 ) ^{2}}{9} = 1

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