What is (x^2)/9+(2x)/3+16y^2-64y+569/9 =0 ?

The solution to (x^2)/9+(2x)/3+16y^2-64y+569/9 =0 is Ellipse with (h,k)=(-3,2),a=4,b= 1/3

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(x^2)/9+(2x)/3+16y^2-64y+569/9 =0

\frac{x ^{2}}{9} + \frac{2 x}{3} + 16 y^{2} - 64 y + \frac{569}{9} = 0

(h, k) = (- 3, 2), a = 4, b = \frac{1}{3}

Solution steps

\frac{x ^{2}}{9} + \frac{2 x}{3} + 16 y^{2} - 64 y + \frac{569}{9} = 0

Rewrite

\frac{x ^{2}}{9} + \frac{2 x}{3} + 16 y^{2} - 64 y + \frac{569}{9} = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (- 3, 2), a = 4, b = \frac{1}{3}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (- 3, 2), a = 4, b = \frac{1}{3}

\frac{( x - 2 ) ^{2}}{4} + \frac{( y + 1 ) ^{2}}{16} = 1

\frac{( ( x + 0 ) ^{2} )}{20} + \frac{( ( y + 0 ) ^{2} )}{16} = 1

x^{2} + 4 y^{2} - 4 x - 8 y = 0

x = \frac{x ^{2}}{16} + \frac{z ^{2}}{9}

4 x^{2} + 25 y^{2} + 549 = 72 x + 250 y

What is (x^2)/9+(2x)/3+16y^2-64y+569/9 =0 ?
The solution to (x^2)/9+(2x)/3+16y^2-64y+569/9 =0 is Ellipse with (h,k)=(-3,2),a=4,b= 1/3