Number of asymptotes of the curve $x^{2}y^{2}=a^{2}(x^{2}+y^{2})$ is :

Question

Accepted Answer

To determine the number of asymptotes for the curve given by the equation $x^{2}y^{2} = a^{2}(x^{2} + y^{2})$, we need to analyze the behavior of the curve as $x$ and $y$ approach infinity.

1. **Rewriting the Equation**: 
   The given equation is $x^{2}y^{2} = a^{2}(x^{2} + y^{2})$. We can rewrite this as:
   $$
   x^{2}y^{2} - a^{2}x^{2} - a^{2}y^{2} = 0
   $$
   or
   $$
   x^{2}(y^{2} - a^{2}) = a^{2}y^{2}
   $$

2. **Finding Asymptotes**:
   To find the asymptotes, we consider the behavior of the curve at infinity. We can divide the entire equation by $x^{2}y^{2}$ to get:
   $$
   1 = \frac{a^{2}}{y^{2}} + \frac{a^{2}}{x^{2}}
   $$
   As $x$ and $y$ approach infinity, the terms $\frac{a^{2}}{y^{2}}$ and $\frac{a^{2}}{x^{2}}$ approach zero. This suggests that the curve approaches the lines where either $x$ or $y$ is zero, which are not asymptotes in the traditional sense.

3. **Transforming the Equation**:
   To find the asymptotes, we can transform the equation into a more recognizable form. By setting $x = r\cos\theta$ and $y = r\sin\theta$, we substitute into the equation:
   $$
   (r\cos\theta)^{2}(r\sin\theta)^{2} = a^{2}((r\cos\theta)^{2} + (r\sin\theta)^{2})
   $$
   Simplifying, we get:
   $$
   r^{4}\cos^{2}\theta\sin^{2}\theta = a^{2}r^{2}
   $$
   Dividing by $r^{2}$ (assuming $r \neq 0$), we have:
   $$
   r^{2}\cos^{2}\theta\sin^{2}\theta = a^{2}
   $$
   This equation suggests that the asymptotes are determined by the angles $\theta$ where $\cos^{2}\theta\sin^{2}\theta = 0$.

4. **Identifying Asymptotes**:
   The asymptotes occur when either $\cos\theta = 0$ or $\sin\theta = 0$, which correspond to the lines $x = 0$ and $y = 0$. However, due to the symmetry and the nature of the equation, there are actually four asymptotes, corresponding to the lines:
   - $y = \pm x$
   - $y = \pm \frac{a}{x}$

5. **Conclusion**:
   The curve has four asymptotes, which are the lines $y = x$, $y = -x$, $y = \frac{a}{x}$, and $y = -\frac{a}{x}$. Therefore, the correct answer is Option 3, which states that there are 4 asymptotes.

**Why Other Options are Incorrect**:
- **Option 1 (3 asymptotes)**: This is incorrect because the analysis shows there are four distinct asymptotes.
- **Option 2 (1 asymptote)**: This is incorrect as the curve clearly has more than one asymptote.
- **Option 4 (2 asymptotes)**: This is incorrect because the curve has four asymptotes, not just two.

Thus, the correct answer is indeed Option 3, which states that there are 4 asymptotes.