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What is the center of the hyperbola $x^2 - y^2 - 100 = 0$ ? $\newline$ $(\_,\_)$

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Find properties of hyperbolas from equations in general form

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Q. What is the center of the hyperbola

x^2 - y^2 - 100 = 0

?

\newline

(\_,\_)

Rewrite Equation: Rewrite the given hyperbola equation in standard form. $\newline$ The given equation is $x^2 - y^2 - 100 = 0$ . To find the center, we need to express the equation in the standard form of a hyperbola, which is $(x-h)^2/a^2 - (y-k)^2/b^2 = 1$ , where $(h, k)$ is the center of the hyperbola. $\newline$ First, we move the constant term to the right side of the equation: $\newline$ $x^2 - y^2 = 100$
Divide by Constant: Divide the equation by the constant term on the right side to get the standard form. $\newline$ Dividing both sides by $100$ , we get: $\newline$ $\frac{x^2}{100} - \frac{y^2}{100} = 1$ $\newline$ This can be rewritten as: $\newline$ $\frac{x^2}{10^2} - \frac{y^2}{10^2} = 1$
Identify Center: Identify the center of the hyperbola. The equation is now in the form $(\frac{x^2}{10^2}) - (\frac{y^2}{10^2}) = 1$ , which is similar to the standard form $(\frac{(x-h)^2}{a^2}) - (\frac{(y-k)^2}{b^2}) = 1$ . Since there are no terms $(x-h)$ or $(y-k)$ , we can conclude that $h = 0$ and $k = 0$ . Therefore, the center of the hyperbola is at the point $(h, k) = (0, 0)$ .

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