1. **Problem 1a:** Write the equation of the circle with center at (-2, 3) passing through (3, 7). 2. The formula for a circle with center $(h,k)$ and radius $r$ is: $$ (x - h)^2 + (y - k)^2 = r^2 $$ 3. First, find the radius $r$ by calculating the distance between the center $(-2,3)$ and the point $(3,7)$: $$ r = \sqrt{(3 - (-2))^2 + (7 - 3)^2} = \sqrt{(3 + 2)^2 + 4^2} = \sqrt{5^2 + 4^2} = \sqrt{25 + 16} = \sqrt{41} $$ 4. Substitute $h = -2$, $k = 3$, and $r^2 = 41$ into the standard form: $$ (x + 2)^2 + (y - 3)^2 = 41 $$ 5. To write the general form, expand the standard form: $$ (x + 2)^2 + (y - 3)^2 = 41 $$ $$ x^2 + 4x + 4 + y^2 - 6y + 9 = 41 $$ $$ x^2 + y^2 + 4x - 6y + (4 + 9 - 41) = 0 $$ $$ x^2 + y^2 + 4x - 6y - 28 = 0 $$ --- 2. **Problem 2a:** Express the equation $x^2 + y^2 - 10x - 14y - 7 = 0$ in standard form and find the center and radius. 3. Group $x$ and $y$ terms and complete the square: $$ x^2 - 10x + y^2 - 14y = 7 $$ 4. Complete the square for $x$: $$ x^2 - 10x = (x^2 - 10x + 25) - 25 = (x - 5)^2 - 25 $$ 5. Complete the square for $y$: $$ y^2 - 14y = (y^2 - 14y + 49) - 49 = (y - 7)^2 - 49 $$ 6. Substitute back: $$ (x - 5)^2 - 25 + (y - 7)^2 - 49 = 7 $$ 7. Simplify: $$ (x - 5)^2 + (y - 7)^2 - 74 = 7 $$ $$ (x - 5)^2 + (y - 7)^2 = 7 + 74 = 81 $$ 8. The center is $(5,7)$ and the radius is $r = \sqrt{81} = 9$. --- **Final answers:** 1a. Standard form: $$(x + 2)^2 + (y - 3)^2 = 41$$ General form: $$x^2 + y^2 + 4x - 6y - 28 = 0$$ 2a. Standard form: $$(x - 5)^2 + (y - 7)^2 = 81$$ Center: $$(5,7)$$ Radius: $$9$$