What is 12x^2+4y^2-24x-4y+1=0 ?

The solution to 12x^2+4y^2-24x-4y+1=0 is Ellipse with (h,k)=(1, 1/2),a=1,b=sqrt(3)

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12x^2+4y^2-24x-4y+1=0

12 x^{2} + 4 y^{2} - 24 x - 4 y + 1 = 0

(h, k) = (1, \frac{1}{2}), a = 1, b = 3

Solution steps

12 x^{2} + 4 y^{2} - 24 x - 4 y + 1 = 0

Rewrite

12 x^{2} + 4 y^{2} - 24 x - 4 y + 1 = 0

in the form of the standard ellipse equation

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

Therefore ellipse properties are:

(h, k) = (1, \frac{1}{2}), a = 1, b = 3

b > a

therefore

b

is semi-major axis and

a

is semi-minor axis

Ellipse with center (h, k) = (1, \frac{1}{2}), b = 3, a = 1

\frac{( x - 2 ) ^{2}}{25} + \frac{( y + 1 ) ^{2}}{9} = 1

4 x^{2} + 5 y^{2} = 20

\frac{( x - 2 ) ^{2}}{36} + \frac{( y - 3 ) ^{2}}{81} = 1

36 x^{2} + 16 y^{2} - 180 x + 96 y - 207 = 0

x^{2} + 3 y^{2} = 9

What is 12x^2+4y^2-24x-4y+1=0 ?
The solution to 12x^2+4y^2-24x-4y+1=0 is Ellipse with (h,k)=(1, 1/2),a=1,b=sqrt(3)