1. **State the problem:** We need to analyze and rewrite the equation $$9x^2 + 9y^2 - 6x + 18y + 11 = 0$$ to identify its geometric shape and properties. 2. **Rewrite the equation:** Divide the entire equation by 9 to simplify coefficients: $$x^2 + y^2 - \frac{2}{3}x + 2y + \frac{11}{9} = 0$$ 3. **Group x and y terms:** $$\left(x^2 - \frac{2}{3}x\right) + \left(y^2 + 2y\right) = -\frac{11}{9}$$ 4. **Complete the square:** - For $x$: Take half of $-\frac{2}{3}$, which is $-\frac{1}{3}$, square it to get $\frac{1}{9}$. - For $y$: Take half of $2$, which is $1$, square it to get $1$. Add these squares to both sides: $$\left(x^2 - \frac{2}{3}x + \frac{1}{9}\right) + \left(y^2 + 2y + 1\right) = -\frac{11}{9} + \frac{1}{9} + 1$$ 5. **Simplify:** $$\left(x - \frac{1}{3}\right)^2 + (y + 1)^2 = -\frac{11}{9} + \frac{1}{9} + 1 = -\frac{11}{9} + \frac{1}{9} + \frac{9}{9} = -\frac{11}{9} + \frac{10}{9} = -\frac{1}{9}$$ 6. **Interpretation:** The right side is negative ($-\frac{1}{9}$), which means the equation does not represent a real circle because the radius squared cannot be negative. **Final answer:** The equation does not represent a real circle or any real locus since the radius squared is negative.