What is 9x^{(2)}-54x+y^{(2)}-2y=-81 ?

The solution to 9x^{(2)}-54x+y^{(2)}-2y=-81 is Ellipse with (h,k)=(3,1),a= 1/3 ,b=1

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9x^{(2)}-54x+y^{(2)}-2y=-81

9 x^{(2)} - 54 x + y^{(2)} - 2 y = - 81

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

Solution steps

9 x^{2} - 54 x + y^{2} - 2 y = - 81

Rewrite

9 x^{2} - 54 x + y^{2} - 2 y = - 81

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

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

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

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

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

4 x^{2} + z^{2} = 36

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

16 x^{2} + 4 y^{2} = 49

4 x^{2} + 8 y^{2} - 16 = 0

What is 9x^{(2)}-54x+y^{(2)}-2y=-81 ?
The solution to 9x^{(2)}-54x+y^{(2)}-2y=-81 is Ellipse with (h,k)=(3,1),a= 1/3 ,b=1