What is 3x^2+2y^2-12x+8y+19=0 ?

The solution to 3x^2+2y^2-12x+8y+19=0 is Ellipse with (h,k)=(2,-2),a=sqrt(1/3),b=sqrt(1/2)

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

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

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

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

\frac{( x + 5 ) ^{2}}{16} + \frac{( y - 3 ) ^{2}}{25} = 1

\frac{y ^{2}}{16} + \frac{x ^{2}}{9} = 1

0.25 x^{2} + y^{2} = 1

5 (x - 3)^{2} = 20 - 4 (y - 6)^{2}

What is 3x^2+2y^2-12x+8y+19=0 ?
The solution to 3x^2+2y^2-12x+8y+19=0 is Ellipse with (h,k)=(2,-2),a=sqrt(1/3),b=sqrt(1/2)