What is (y-4)^2=(4(x+1)^2+84)/3 ?

The solution to (y-4)^2=(4(x+1)^2+84)/3 is Hyperbola with (h,k)=(-1,4),a=2sqrt(7),b=sqrt(21)

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(y-4)^2=(4(x+1)^2+84)/3

(y - 4)^{2} = \frac{4 ( x + 1 ) ^{2} + 84}{3}

(h, k) = (- 1, 4), a = 27, b = 21

Solution steps

(y - 4)^{2} = \frac{4 ( x + 1 ) ^{2} + 84}{3}

Rewrite

(y - 4)^{2} - \frac{4 ( x + 1 ) ^{2} + 84}{3} = 0

in the form of a standard hyperbola equation

\frac{( y - 4 ) ^{2}}{( 2 7 ) ^{2}} - \frac{( x - ( - 1 ) ) ^{2}}{( 21 ) ^{2}} = 1

Therefore Hyperbola properties are:

(h, k) = (- 1, 4), a = 27, b = 21

3 (x + \frac{1}{3})^{2} - 3 (y - \frac{7}{3})^{2} = - 40

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

\frac{( y - 3 ) ^{2}}{169} - \frac{( x + 2 ) ^{2}}{81} = 1

4 y^{2} - 9 x^{2} = 1

x^{2} - \frac{1}{2} y^{2} - \frac{1}{2} y = 0

What is (y-4)^2=(4(x+1)^2+84)/3 ?
The solution to (y-4)^2=(4(x+1)^2+84)/3 is Hyperbola with (h,k)=(-1,4),a=2sqrt(7),b=sqrt(21)