Step 1: Conceptual Definition of Asymptotes:
Asymptotes are lines that a curve approaches as it tends towards infinity. Horizontal asymptotes are determined by evaluating the function's limit as \(x \to \pm\infty\). Vertical asymptotes are found by identifying the x-values where the function's output approaches \(y \to \pm\infty\).
Step 2: Method of Analysis:
The initial step involves rearranging the equation to express one variable in terms of the other, enabling analysis of its behavior at extreme values.
The given equation is: \( (x^2 - a^2)(y^2 - b^2) = a^2b^2 \)
Expanding this yields: \( x^2y^2 - b^2x^2 - a^2y^2 + a^2b^2 = a^2b^2 \)
Simplifying the equation: \[ x^2y^2 - b^2x^2 - a^2y^2 = 0 \]
Step 3: Detailed Derivations:
1. Determination of Horizontal Asymptotes:
To ascertain the horizontal asymptotes, we examine the behavior of \(y\) as \(x\) approaches positive or negative infinity.
The equation is rearranged to isolate \(y^2\):
\[ y^2(x^2 - a^2) = b^2x^2 \]
\[ y^2 = \frac{b^2x^2}{x^2 - a^2} \]
Now, we compute the limit as \(x \to \infty\):
\[ \lim_{x\to\infty} y^2 = \lim_{x\to\infty} \frac{b^2x^2}{x^2 - a^2} = \lim_{x\to\infty} \frac{b^2}{1 - a^2/x^2} = \frac{b^2}{1-0} = b^2 \]
This indicates that as \(x \to \infty\), \(y^2 \to b^2\), which implies \(y \to \pm b\).
Consequently, the horizontal asymptotes are \(y = b\) and \(y = -b\).
2. Determination of Vertical Asymptotes:
Vertical asymptotes are found by considering the behavior of \(x\) as \(y \to \pm\infty\). Alternatively, we can identify the x-values that cause the denominator in the expression for \(y^2\) to be zero, leading to an infinite value for \(y\).
The denominator is \(x^2 - a^2\). Setting this to zero:
\[ x^2 - a^2 = 0 \implies x^2 = a^2 \implies x = \pm a \]
Thus, the vertical asymptotes are \(x = a\) and \(x = -a\).
Step 4: Conclusion:
The asymptotes for the given curve are the lines \(x = a\), \(x = -a\), \(y = b\), and \(y = -b\). This can be concisely expressed as \(x = \pm a\) and \(y = \pm b\).