1. Let's start by stating the problem: We want to understand why a given equation represents a hyperbola. 2. The general form of a hyperbola centered at the origin is given by the equation $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ or $$-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ where $a$ and $b$ are real numbers. 3. Important rule: A hyperbola is characterized by the difference of squares in the equation, meaning one term is positive and the other is negative, and the equation equals 1. 4. To identify if an equation is a hyperbola, rewrite it in the form $$Ax^2 + By^2 + C = 0$$ and check the signs of $A$ and $B$. 5. If $A$ and $B$ have opposite signs (one positive, one negative), the conic is a hyperbola. 6. For example, if the equation is $$9x^2 - 16y^2 = 144$$, dividing both sides by 144 gives $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ which matches the hyperbola form. 7. Therefore, the key is the presence of a difference of squared terms equal to 1, indicating a hyperbola. Final answer: The equation represents a hyperbola because it can be rewritten in the form $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ where the $x^2$ and $y^2$ terms have opposite signs, a defining property of hyperbolas.