1. **Stating the problem:** We are given the equation $$x^a y^b (x^2 - y^2)^2 = (x^2 + y^2)^3$$ and asked to find the inclined asymptotes. 2. **Understanding the problem:** Inclined asymptotes occur when the curve behaves like a line at infinity. We analyze the behavior of the equation as $x$ and $y$ become very large. 3. **Rewrite the equation:** The equation is $$x^a y^b (x^2 - y^2)^2 = (x^2 + y^2)^3$$. 4. **Consider the dominant terms at infinity:** For large $x$ and $y$, $x^2 + y^2$ dominates, so the right side behaves like $(x^2 + y^2)^3$. 5. **Express $y$ in terms of $x$ for asymptotes:** Let $y = mx$, where $m$ is the slope of the asymptote. 6. **Substitute $y = mx$ into the equation:** $$x^a (mx)^b (x^2 - (mx)^2)^2 = (x^2 + (mx)^2)^3$$ Simplify powers of $x$: $$x^a x^b m^b (x^2 - m^2 x^2)^2 = (x^2 + m^2 x^2)^3$$ $$x^{a+b} m^b (x^2(1 - m^2))^2 = (x^2(1 + m^2))^3$$ $$x^{a+b} m^b x^4 (1 - m^2)^2 = x^6 (1 + m^2)^3$$ 7. **Combine powers of $x$:** $$x^{a+b+4} m^b (1 - m^2)^2 = x^6 (1 + m^2)^3$$ 8. **Equate powers of $x$ for balance:** $$a + b + 4 = 6 \\ a + b = 2$$ 9. **Equate coefficients:** $$m^b (1 - m^2)^2 = (1 + m^2)^3$$ 10. **Solve for $m$:** This is the key equation for the slopes of the inclined asymptotes: $$m^b (1 - m^2)^2 = (1 + m^2)^3$$ 11. **Interpretation:** The values of $m$ satisfying this equation give the slopes of the inclined asymptotes. **Final answer:** The inclined asymptotes have slopes $m$ satisfying $$m^b (1 - m^2)^2 = (1 + m^2)^3$$ with the condition on exponents $a + b = 2$ for the powers of $x$ to balance. This completes the analysis of the inclined asymptotes for the given curve.