To find the equation of a circle in general form $Ax^2 + Ay^2 + Dx + Ey + F = 0$, we can start from the standard form of the circle equation $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius.

Problem 4:

A circle with the radius of $2\sqrt{17}$ units with its center $(0, -1)$

1. Standard Form: $(x - 0)^2 + (y + 1)^2 = (2\sqrt{17})^2$ Simplifying: $x^2 + (y + 1)^2 = 4 \cdot 17$ $x^2 + (y + 1)^2 = 68$

2. Expanding and Simplifying: $x^2 + y^2 + 2y + 1 = 68$ $x^2 + y^2 + 2y + 1 - 68 = 0$ $x^2 + y^2 + 2y - 67 = 0$

General Form: $x^2 + y^2 + 2y - 67 = 0$

Problem 5:

A circle whose center is at $(6, -8)$ with a diameter that is $6\sqrt{13}$ units

1. Radius Calculation: $r = \frac{6\sqrt{13}}{2} = 3\sqrt{13}$

2. Standard Form: $(x - 6)^2 + (y + 8)^2 = (3\sqrt{13})^2$ Simplifying: $(x - 6)^2 + (y + 8)^2 = 9 \cdot 13$ $(x - 6)^2 + (y + 8)^2 = 117$

3. Expanding and Simplifying: $(x - 6)^2 + (y + 8)^2 = 117$ $x^2 - 12x + 36 + y^2 + 16y + 64 = 117$ $x^2 + y^2 - 12x + 16y + 100 - 117 = 0$ $x^2 + y^2 - 12x + 16y - 17 = 0$

General Form: $x^2 + y^2 - 12x + 16y - 17 = 0$

Problem 6:

A circle whose center is at $(5,0)$ and passes through the point $(9,3)$

1. Radius Calculation: Using the distance formula: $r = \sqrt{(9 - 5)^2 + (3 - 0)^2}$ $r = \sqrt{4^2 + 3^2}$ $r = \sqrt{16 + 9}$ $r = \sqrt{25}$ $r = 5$

2. Standard Form: $(x - 5)^2 + (y - 0)^2 = 5^2$ Simplifying: $(x - 5)^2 + y^2 = 25$

3. Expanding and Simplifying: $(x - 5)^2 + y^2 = 25$ $x^2 - 10x + 25 + y^2 = 25$ $x^2 + y^2 - 10x + 25 - 25 = 0$ $x^2 + y^2 - 10x = 0$

General Form: $x^2 + y^2 - 10x = 0$

Do you want more details or have any questions?

Here are some related questions to explore further:

1. How do you derive the general form from the standard form of a circle?
2. What are the key differences between the general and standard forms of a circle's equation?
3. How can you find the center and radius from a circle's general equation?
4. How do you calculate the radius of a circle given the diameter?
5. What is the distance formula used for in circle problems?
6. How can you determine if a point lies on a circle given its equation?
7. What steps are involved in converting a circle's equation from standard to general form?
8. How does changing the center of a circle affect its equation?

Tip: Always verify your final circle equation by plugging in the center coordinates and radius back into the standard form to check for consistency.