Question

find the center of this hyperbola.
x² - 25y² - 14x + 100y - 76 = 0
center : ( ?, )

Explanation:

Step1: Group x - terms and y - terms

$(x^{2}-14x)-25(y^{2} - 4y)-76=0$

Step2: Complete the square for x - terms

For $x^{2}-14x$, we have $(x - 7)^{2}-49$.
For $y^{2}-4y$, we have $(y - 2)^{2}-4$.
The equation becomes $(x - 7)^{2}-49-25((y - 2)^{2}-4)-76 = 0$.

Step3: Simplify the equation

$(x - 7)^{2}-49-25(y - 2)^{2}+100 - 76=0$.
$(x - 7)^{2}-25(y - 2)^{2}=25$.
The standard - form of a hyperbola is $\frac{(x - h)^{2}}{a^{2}}-\frac{(y - k)^{2}}{b^{2}} = 1$ (for a hyperbola opening left - right) or $\frac{(y - k)^{2}}{a^{2}}-\frac{(x - h)^{2}}{b^{2}} = 1$ (for a hyperbola opening up - down), and the center is $(h,k)$.
Here, $h = 7$ and $k = 2$.

Answer:

$(7,2)$