1. **State the problem:** Convert the rectangular equation $$21x = y^2$$ to polar form and solve for $$r$$ in terms of $$\theta$$. 2. **Recall the polar-rectangular relationships:** $$x = r \cos\theta$$ $$y = r \sin\theta$$ 3. **Substitute these into the equation:** $$21x = y^2 \implies 21(r \cos\theta) = (r \sin\theta)^2$$ 4. **Simplify the right side:** $$21r \cos\theta = r^2 \sin^2\theta$$ 5. **Rearrange to isolate terms:** $$r^2 \sin^2\theta - 21r \cos\theta = 0$$ 6. **Factor out $$r$$:** $$r(r \sin^2\theta - 21 \cos\theta) = 0$$ 7. **Set each factor equal to zero:** - $$r = 0$$ (trivial solution) - $$r \sin^2\theta - 21 \cos\theta = 0$$ 8. **Solve for $$r$$:** $$r \sin^2\theta = 21 \cos\theta$$ $$r = \frac{21 \cos\theta}{\sin^2\theta}$$ 9. **Final polar form solving for $$r$$:** $$\boxed{r = \frac{21 \cos\theta}{\sin^2\theta}}$$ This expresses $$r$$ in terms of $$\theta$$ for the given equation.