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

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

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

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

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

Solution steps

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

Rewrite

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

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 ^{2}}{b ^{2}} + \frac{y ^{2}}{a ^{2}} = 1

4 x^{2} + 25 y^{2} - 100 = 0

16 x^{2} + 25 y^{2} = 100

64 x^{2} + 100 y^{2} + 768 x - 600 y - 3196 = 0

x^{2} + 4 y^{2} + 6 x - 8 y + 9 = 0

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