What is 4x^2+9y^2-54y+45=0 ?

The solution to 4x^2+9y^2-54y+45=0 is Ellipse with (h,k)=(0,3),a=3,b=2

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4x^2+9y^2-54y+45=0

4 x^{2} + 9 y^{2} - 54 y + 45 = 0

(h, k) = (0, 3), a = 3, b = 2

Solution steps

4 x^{2} + 9 y^{2} - 54 y + 45 = 0

Rewrite

4 x^{2} + 9 y^{2} - 54 y + 45 = 0

in the form of the standard ellipse equation

\frac{( x - 0 ) ^{2}}{3 ^{2}} + \frac{( y - 3 ) ^{2}}{2 ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (0, 3), a = 3, b = 2

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (0, 3), a = 3, b = 2

x^{2} + 4 y^{2} = 5

3 x^{2} + 4 y^{2} + 18 x + 8 y + 19 = 0

\frac{( x + 3 ) ^{2}}{4} + \frac{( y + 2 ) ^{2}}{16} = 1

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

\frac{- x ^{2}}{2 ^{2}} - \frac{y ^{2}}{4 ^{2}} = 1

What is 4x^2+9y^2-54y+45=0 ?
The solution to 4x^2+9y^2-54y+45=0 is Ellipse with (h,k)=(0,3),a=3,b=2