Part 4/5:
[ 9x^2 - 4y^2 - 72x + 8y + 176 = 0 ]
First, we rearrange:
[ 9x^2 - 72x - 4y^2 + 8y = -176 ]
Completing the square for both the x and y terms enables us to rewrite the equation into the standard form of a hyperbola. Adding and subtracting necessary terms, we arrive at a standard hyperbola after simplification:
[ 4(y - 1)^2 - 9(x - 4)^2 = 36 ]
Dividing by 36 gives:
[ \frac{(y - 1)^2}{9} - \frac{(x - 4)^2}{4} = 1 ]
Here, the center of the hyperbola is shifted to (4, 1), with vertices and asymptotes recalculated based on the transformations we applied.