1. **Write the standard form equation of the circle from** $137 + 6y = -y^2 - x^2 - 24x$. 2. **Rewrite the equation by moving all terms to one side:** $$x^2 + y^2 + 24x + 6y + 137 = 0$$ 3. **Group $x$ and $y$ terms:** $$x^2 + 24x + y^2 + 6y = -137$$ 4. **Complete the square for $x$ and $y$ terms:** - For $x^2 + 24x$, half of 24 is 12, square is $12^2 = 144$. - For $y^2 + 6y$, half of 6 is 3, square is $3^2 = 9$. 5. **Add these squares to both sides:** $$x^2 + 24x + 144 + y^2 + 6y + 9 = -137 + 144 + 9$$ 6. **Simplify:** $$(x + 12)^2 + (y + 3)^2 = 16$$ 7. **Final standard form:** $$\boxed{(x + 12)^2 + (y + 3)^2 = 16}$$ --- 2. **Given:** $x^2 + y^2 + 14x - 12y + 4 = 0$ 3. **Group $x$ and $y$ terms:** $$x^2 + 14x + y^2 - 12y = -4$$ 4. **Complete the square:** - For $x^2 + 14x$, half of 14 is 7, square is $7^2 = 49$. - For $y^2 - 12y$, half of -12 is -6, square is $(-6)^2 = 36$. 5. **Add to both sides:** $$x^2 + 14x + 49 + y^2 - 12y + 36 = -4 + 49 + 36$$ 6. **Simplify:** $$(x + 7)^2 + (y - 6)^2 = 81$$ 7. **Final standard form:** $$\boxed{(x + 7)^2 + (y - 6)^2 = 81}$$ --- 3. **Ends of diameter:** $(-3, 11)$ and $(3, -13)$ 4. **Find center (midpoint):** $$\left(\frac{-3 + 3}{2}, \frac{11 + (-13)}{2}\right) = (0, -1)$$ 5. **Find radius (half the distance between points):** $$d = \sqrt{(3 - (-3))^2 + (-13 - 11)^2} = \sqrt{6^2 + (-24)^2} = \sqrt{36 + 576} = \sqrt{612} = 6\sqrt{17}$$ 6. **Radius:** $$r = \frac{d}{2} = 3\sqrt{17}$$ 7. **Standard form:** $$\boxed{(x - 0)^2 + (y + 1)^2 = (3\sqrt{17})^2 = 9 \times 17 = 153}$$ --- 4. **Center:** $(2, -5)$, **Point on circle:** $(-7, -1)$ 5. **Find radius (distance from center to point):** $$r = \sqrt{(-7 - 2)^2 + (-1 + 5)^2} = \sqrt{(-9)^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97}$$ 6. **Standard form:** $$\boxed{(x - 2)^2 + (y + 5)^2 = 97}$$ --- 5. **Circle centered at origin with radius 3:** 6. **Standard form:** $$\boxed{x^2 + y^2 = 9}$$