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

The solution to 9x^2-4y^2+54x-8y+78=0 is Hyperbola with (h,k)=(-3,-1),a= 1/2 ,b= 1/3

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

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

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

Solution steps

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

Rewrite

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

in the form of a standard hyperbola equation

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

Therefore Hyperbola properties are:

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

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

8 x^{2} - 3 y^{2} = 120

5 y^{2} - 16 x^{2} + 400 = 0

9 x^{2} - 4 y^{2} + 54 x + 8 y + 41 = 0

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

What is 9x^2-4y^2+54x-8y+78=0 ?
The solution to 9x^2-4y^2+54x-8y+78=0 is Hyperbola with (h,k)=(-3,-1),a= 1/2 ,b= 1/3