What is 27x^2-54x+48y^2-192y-213=0 ?

The solution to 27x^2-54x+48y^2-192y-213=0 is Ellipse with (h,k)=(1,2),a=4,b=3

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27x^2-54x+48y^2-192y-213=0

27 x^{2} - 54 x + 48 y^{2} - 192 y - 213 = 0

(h, k) = (1, 2), a = 4, b = 3

Solution steps

27 x^{2} - 54 x + 48 y^{2} - 192 y - 213 = 0

Rewrite

27 x^{2} - 54 x + 48 y^{2} - 192 y - 213 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (1, 2), a = 4, b = 3

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (1, 2), a = 4, b = 3

\frac{( x - 5 ) ^{2}}{4} + \frac{( y - 4 ) ^{2}}{16} = 1

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

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

9 x^{(2)} + 4 y^{(2)} + 36 x - 24 y + 36 = 0

9 (y^{2} + 6 y + 9) + 16 (x^{2} - 4 x + 4) = 81 + 64

What is 27x^2-54x+48y^2-192y-213=0 ?
The solution to 27x^2-54x+48y^2-192y-213=0 is Ellipse with (h,k)=(1,2),a=4,b=3