What is the foci x^2+1/3 y^2=x+y ?

The foci x^2+1/3 y^2=x+y is (1/2 ,(3+2sqrt(2))/2),(1/2 ,(3-2sqrt(2))/2)

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foci x^2+1/3 y^2=x+y

foci

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

(\frac{1}{2}, \frac{3 + 2 2}{2}), (\frac{1}{2}, \frac{3 - 2 2}{2})

Solution steps

(h, k + c), (h, k - c)

Calculate ellipse properties

x^{2} + \frac{1}{3} y^{2} = x + y :

Ellipse with center

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

(\frac{1}{2}, \frac{3}{2} + c), (\frac{1}{2}, \frac{3}{2} - c)

Compute

c :

c = 3^{2} - 1^{2} : 2

(\frac{1}{2}, \frac{3}{2} + 2), (\frac{1}{2}, \frac{3}{2} - 2)

Refine

(\frac{1}{2}, \frac{3 + 2 2}{2}), (\frac{1}{2}, \frac{3 - 2 2}{2})

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

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

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

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

9 x^{2} + 36 y^{2} + 36 x + 432 y = - 1008

What is the foci x^2+1/3 y^2=x+y ?
The foci x^2+1/3 y^2=x+y is (1/2 ,(3+2sqrt(2))/2),(1/2 ,(3-2sqrt(2))/2)