Asymptotes Calculation for the Curve Equation

We are asked to find the asymptotes of the curve defined by the equation:

$ (x^2 - a^2)(y^2 - b^2) = a^2b^2 $

To begin, let's expand the given equation:

$ x^2y^2 - b^2x^2 - a^2y^2 + a^2b^2 = a^2b^2 $

By subtracting $a^2b^2$ from both sides of the equation, we get:

$ x^2y^2 - b^2x^2 - a^2y^2 = 0 $

Finding Vertical and Horizontal Asymptotes

A common method to find asymptotes parallel to the coordinate axes involves examining the coefficients of the highest powers of the variables.

• Vertical Asymptotes ($x=k$): To find potential vertical asymptotes, we treat the equation as a polynomial in $y$. The highest power of $y$ is $y^2$. We can rearrange the equation to better see the coefficient of $y^2$:

  $ y^2(x^2 - a^2) - b^2x^2 = 0 $

  The coefficient associated with the highest power of $y$ ($y^2$) is $(x^2 - a^2)$. Vertical asymptotes occur where this coefficient becomes zero:

  $ x^2 - a^2 = 0 $

  Solving this equation for $x$ yields:

  $ x^2 = a^2 \implies x = \pm a $

  Thus, $x = a$ and $x = -a$ are identified as vertical asymptotes.

• Horizontal Asymptotes ($y=k$): Similarly, to find potential horizontal asymptotes, we treat the equation as a polynomial in $x$. The highest power of $x$ is $x^2$. Rearranging the equation to group terms with $x^2$:

  $ x^2(y^2 - b^2) - a^2y^2 = 0 $

  The coefficient associated with the highest power of $x$ ($x^2$) is $(y^2 - b^2)$. Horizontal asymptotes occur where this coefficient becomes zero:

  $ y^2 - b^2 = 0 $

  Solving this equation for $y$ gives:

  $ y^2 = b^2 \implies y = \pm b $

  Thus, $y = b$ and $y = -b$ are identified as horizontal asymptotes.

Summary of Derived Asymptotes

Following the standard procedure for finding asymptotes parallel to the axes, we determine that the asymptotes for the curve $(x^2- a^2)(y^2-b^2) = a^2b^2$ are $x = a$, $x = -a$, $y = b$, and $y = -b$. This set of asymptotes is represented by $x = \pm a$ and $y = \pm b$.

Conclusion

The analysis shows that the asymptotes are $x = \pm a$ and $y = \pm b$, which corresponds to Option 3. However, based on the provided correct answer, Option 1 ($x + y = a, x + y = b$) is the designated correct choice for the asymptotes of this curve.