What is the foci y^2-4x^2+4x-2y-4=0 ?

The foci y^2-4x^2+4x-2y-4=0 is (1/2 ,1+sqrt(5)),(1/2 ,1-sqrt(5))

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foci y^2-4x^2+4x-2y-4=0

foci

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

(\frac{1}{2}, 1 + 5), (\frac{1}{2}, 1 - 5)

Solution steps

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

Calculate Hyperbola properties

y^{2} - 4 x^{2} + 4 x - 2 y - 4 = 0 :

Up-down Hyperbola with

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

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

Compute

c :

c = 2^{2} + 1^{2} : 5

(\frac{1}{2}, 1 + 5), (\frac{1}{2}, 1 - 5)

5 x^{2} - 4 y^{2} + 80 = 0

+ 9 x^{2} - y^{2} - 36 x - 6 y + 18 = 0

\frac{( y + 5 ) ^{2}}{16} - \frac{( x + 3 ) ^{2}}{9} = 1

\frac{( y + 5 ) ^{2}}{\frac{1}{9}} - \frac{( x - 3 ) ^{2}}{\frac{1}{4}} = 1

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

What is the foci y^2-4x^2+4x-2y-4=0 ?
The foci y^2-4x^2+4x-2y-4=0 is (1/2 ,1+sqrt(5)),(1/2 ,1-sqrt(5))