1. **Stating the problem:** We want to understand the two common forms of the equation of a circle and see examples of each. 2. **Form 1: Standard form of a circle's equation** The standard form is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ where $(h, k)$ is the center of the circle and $r$ is the radius. 3. **Example 1:** Given: $$ (x - 3)^2 + (y - 5)^2 = 9 $$ Here, the center is $(3, 5)$ and the radius is $r = \sqrt{9} = 3$. 4. **Form 2: General form of a circle's equation** The general form is: $$ x^2 + y^2 + Dx + Ey + F = 0 $$ where $D$, $E$, and $F$ are constants. 5. **To convert general form to standard form, complete the square:** 6. **Example 2:** Given: $$ x^2 + y^2 + 3x + 8y = 9 $$ Rewrite as: $$ x^2 + 3x + y^2 + 8y = 9 $$ 7. **Complete the square for $x$ and $y$ terms:** For $x$: $$ x^2 + 3x = x^2 + 3x + \left(\frac{3}{2}\right)^2 - \left(\frac{3}{2}\right)^2 = (x + \frac{3}{2})^2 - \frac{9}{4} $$ For $y$: $$ y^2 + 8y = y^2 + 8y + 16 - 16 = (y + 4)^2 - 16 $$ 8. **Substitute back:** $$ (x + \frac{3}{2})^2 - \frac{9}{4} + (y + 4)^2 - 16 = 9 $$ 9. **Simplify constants:** $$ (x + \frac{3}{2})^2 + (y + 4)^2 = 9 + \frac{9}{4} + 16 $$ Calculate right side: $$ 9 + \frac{9}{4} + 16 = \frac{36}{4} + \frac{9}{4} + \frac{64}{4} = \frac{109}{4} $$ 10. **Final standard form:** $$ (x + \frac{3}{2})^2 + (y + 4)^2 = \frac{109}{4} $$ Center is $\left(-\frac{3}{2}, -4\right)$ and radius is $r = \sqrt{\frac{109}{4}} = \frac{\sqrt{109}}{2}$. **Summary mind map words:** - Standard form: center $(h,k)$, radius $r$ - General form: complete the square - Examples show how to find center and radius