What is the foci 9x^2-4y^2+54x-8y+78=0 ?

The foci 9x^2-4y^2+54x-8y+78=0 is (-3,(-6+sqrt(13))/6),(-3,(-6-sqrt(13))/6)

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foci 9x^2-4y^2+54x-8y+78=0

foci

9 x^{2} - 4 y^{2} + 54 x - 8 y + 78 = 0

(- 3, \frac{- 6 + 13}{6}), (- 3, \frac{- 6 - 13}{6})

Solution steps

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

Calculate Hyperbola properties

9 x^{2} - 4 y^{2} + 54 x - 8 y + 78 = 0 :

Up-down Hyperbola with

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

(- 3, - 1 + c), (- 3, - 1 - c)

Compute

c :

c = (\frac{1}{2})^{2} + (\frac{1}{3})^{2} : \frac{13}{6}

(- 3, - 1 + \frac{13}{6}), (- 3, - 1 - \frac{13}{6})

Refine

(- 3, \frac{- 6 + 13}{6}), (- 3, \frac{- 6 - 13}{6})

9 x^{2} - 36 x - 4 y^{2} - 24 y - 36 = 0

\frac{( x - 3 ) ^{2}}{81} - \frac{( y + 5 ) ^{2}}{144} = 1

\frac{x ^{2}}{1} - \frac{y ^{2}}{9} = 1

y^{2} - x^{2} = 25

\frac{x ^{2}}{140} - \frac{( y - 6 ) ^{2}}{4} = - 1

What is the foci 9x^2-4y^2+54x-8y+78=0 ?
The foci 9x^2-4y^2+54x-8y+78=0 is (-3,(-6+sqrt(13))/6),(-3,(-6-sqrt(13))/6)