What is 6x^2+9y^2-24x-54y+51=0 ?

The solution to 6x^2+9y^2-24x-54y+51=0 is Ellipse with (h,k)=(2,3),a=3,b=sqrt(6)

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6x^2+9y^2-24x-54y+51=0

6 x^{2} + 9 y^{2} - 24 x - 54 y + 51 = 0

(h, k) = (2, 3), a = 3, b = 6

Solution steps

6 x^{2} + 9 y^{2} - 24 x - 54 y + 51 = 0

Rewrite

6 x^{2} + 9 y^{2} - 24 x - 54 y + 51 = 0

in the form of the standard ellipse equation

\frac{( x - 2 ) ^{2}}{3 ^{2}} + \frac{( y - 3 ) ^{2}}{( 6 ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (2, 3), a = 3, b = 6

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (2, 3), a = 3, b = 6

\frac{( x + 3 ) ^{2}}{9} + \frac{( y - 3 ) ^{2}}{16} = 1

\frac{( x - 2 ) ^{2}}{16} + \frac{( y + 3 ) ^{2}}{25} = 1

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

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

1 x^{2} + 3 y^{2} + 9 x + 8 y - 14 = 0

What is 6x^2+9y^2-24x-54y+51=0 ?
The solution to 6x^2+9y^2-24x-54y+51=0 is Ellipse with (h,k)=(2,3),a=3,b=sqrt(6)