Question

The asymptotes of the curve \( (x^2 - a^2)(y^2 - b^2) = a^2b^2 \) are

• A. x + y = a, x + y = b
• B. xy = a, x - y = b
• C. x = \(\pm\)a, y = \(\pm\)b
• D. x = y

Hint:

For algebraic curves, an easy way to find horizontal and vertical asymptotes is to set the coefficients of the highest power of x and y to zero, respectively, after clearing fractions. In \(x^2y^2 - b^2x^2 - a^2y^2 = 0\), the coefficient of \(y^2\) is \(x^2-a^2\). Setting this to zero gives the vertical asymptotes \(x=\pm a\). The coefficient of \(x^2\) is \(y^2-b^2\). Setting this to zero gives the horizontal asymptotes \(y=\pm b\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Asymptotes are lines that a curve approaches as it heads towards infinity. We can find horizontal asymptotes by taking the limit of the function as \(x \to \pm\infty\), and vertical asymptotes by finding the values of x for which the function approaches \(y \to \pm\infty\).
Step 2: Key Formula or Approach:
We first rearrange the equation to express y in terms of x (or vice versa) and then analyze its behavior at infinity. Given equation: \( (x^2 - a^2)(y^2 - b^2) = a^2b^2 \)
Expand the equation: \( x^2y^2 - b^2x^2 - a^2y^2 + a^2b^2 = a^2b^2 \)
\[ x^2y^2 - b^2x^2 - a^2y^2 = 0 \] Step 3: Detailed Explanation:
1. Finding Horizontal Asymptotes:
To find horizontal asymptotes, we analyze the behavior of y as \(x \to \pm\infty\). Rearrange the equation to solve for \(y^2\): \[ y^2(x^2 - a^2) = b^2x^2 \] \[ y^2 = \frac{b^2x^2}{x^2 - a^2} \] Now, take the limit as \(x \to \infty\): \[ \lim_{x\to\infty} y^2 = \lim_{x\to\infty} \frac{b^2x^2}{x^2 - a^2} = \lim_{x\to\infty} \frac{b^2}{1 - a^2/x^2} = \frac{b^2}{1-0} = b^2 \] So, as \(x \to \infty\), \(y^2 \to b^2\), which means \(y \to \pm b\). The horizontal asymptotes are \(y = b\) and \(y = -b\).
2. Finding Vertical Asymptotes:
To find vertical asymptotes, we analyze the behavior of x as \(y \to \pm\infty\). We can also find the values of x for which the denominator in the expression for \(y^2\) becomes zero, causing y to go to infinity. The denominator is \(x^2 - a^2\). Setting it to zero: \[ x^2 - a^2 = 0 \implies x^2 = a^2 \implies x = \pm a \] The vertical asymptotes are \(x = a\) and \(x = -a\).
Step 4: Final Answer:
The asymptotes of the curve are the lines \(x = a\), \(x = -a\), \(y = b\), and \(y = -b\). This can be written as \(x = \pm a, y = \pm b\).