What is the axis 3x^{(2)}+2y^{(2)}-12x+8y+19=0 ?

The axis 3x^{(2)}+2y^{(2)}-12x+8y+19=0 is semi-major axis b=sqrt(1/2), semi-minor axis a=sqrt(1/3)

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axis 3x^{(2)}+2y^{(2)}-12x+8y+19=0

axis

3 x^{(2)} + 2 y^{(2)} - 12 x + 8 y + 19 = 0

semi - major axis b = \frac{1}{2}, semi - minor axis a = \frac{1}{3}

Solution steps

3 x^{2} + 2 y^{2} - 12 x + 8 y + 19 = 0

Rewrite

3 x^{2} + 2 y^{2} - 12 x + 8 y + 19 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

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

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

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

\frac{( x + 2 ) ^{2}}{4} + \frac{( y + 3 ) ^{2}}{9} = 1

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

\frac{( x - 5 ) ^{2}}{9} + \frac{( y - 1 ) ^{2}}{4} = 1

3 x^{2} + 3 y^{2} - 18 y = 0

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

What is the axis 3x^{(2)}+2y^{(2)}-12x+8y+19=0 ?
The axis 3x^{(2)}+2y^{(2)}-12x+8y+19=0 is semi-major axis b=sqrt(1/2), semi-minor axis a=sqrt(1/3)