1. **State the problem:** We are given two equations: $$y = \frac{\sqrt{3x}}{3}$$ and $$x^2 + y^2 = 12cx$$ We want to understand these equations and convert the Cartesian equation to polar coordinates. 2. **Recall the polar coordinate formulas:** In polar coordinates, we use: $$x = r\cos\theta$$ $$y = r\sin\theta$$ where $r$ is the distance from the origin and $\theta$ is the angle from the positive x-axis. 3. **Convert the circle equation to polar form:** Substitute $x = r\cos\theta$ and $y = r\sin\theta$ into $$x^2 + y^2 = 12cx$$ We get: $$r^2 = 12c (r\cos\theta)$$ 4. **Simplify the equation:** $$r^2 = 12cr\cos\theta$$ Divide both sides by $r$ (assuming $r \neq 0$): $$r = 12c\cos\theta$$ 5. **Interpretation:** This is the polar form of the circle equation. It represents a circle with radius depending on $\theta$. 6. **Analyze the first equation:** $$y = \frac{\sqrt{3x}}{3}$$ This can be rewritten as: $$y = \frac{1}{3} \sqrt{3x} = \frac{\sqrt{3}}{3} \sqrt{x}$$ This defines $y$ in terms of $x$ for $x \geq 0$. 7. **Summary:** - The circle equation in polar coordinates is: $$r = 12c\cos\theta$$ - The function $y = \frac{\sqrt{3x}}{3}$ is a curve defined for $x \geq 0$. **Final answer:** $$r = 12c\cos\theta$$