What is the foci 4x^2-24x+(y^2)/(16)+y/8+573/16 =0 ?

The foci 4x^2-24x+(y^2)/(16)+y/8+573/16 =0 is (3,(-4+3sqrt(7))/4),(3,(-4-3sqrt(7))/4)

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foci 4x^2-24x+(y^2)/(16)+y/8+573/16 =0

foci

4 x^{2} - 24 x + \frac{y ^{2}}{16} + \frac{y}{8} + \frac{573}{16} = 0

(3, \frac{- 4 + 3 7}{4}), (3, \frac{- 4 - 3 7}{4})

Solution steps

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

Calculate ellipse properties

4 x^{2} - 24 x + \frac{y ^{2}}{16} + \frac{y}{8} + \frac{573}{16} = 0 :

Ellipse with center

(h, k) = (3, - 1), b = 2, a = \frac{1}{4}

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

Compute

c :

c = 2^{2} - (\frac{1}{4})^{2} : \frac{3 7}{4}

(3, - 1 + \frac{3 7}{4}), (3, - 1 - \frac{3 7}{4})

Refine

(3, \frac{- 4 + 3 7}{4}), (3, \frac{- 4 - 3 7}{4})

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

\frac{( x + 7 ) ^{2}}{16} + \frac{( y + 3 ) ^{2}}{9} = 1

16 x^{2} + 9 y^{2} - 96 x + 36 y - 36 = 0

\frac{x ^{2}}{100} + \frac{y ^{2}}{44} = 1

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

What is the foci 4x^2-24x+(y^2)/(16)+y/8+573/16 =0 ?
The foci 4x^2-24x+(y^2)/(16)+y/8+573/16 =0 is (3,(-4+3sqrt(7))/4),(3,(-4-3sqrt(7))/4)