FIND THE EQUATION OF EACH CIRCLE IN GENERAL FORM, GIVEN THE FOLLOWING INFORMATION.

4. A circle with the radius of 2√17 unites with its center (0,-1)

5. A circle whose center is at (6.-8) with a diameter that is 6√13 units

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

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.

Learn how to find the equation of a circle in general form with detailed examples. Includes circles with given centers and radii or diameters, involving key geometric concepts and the distance formula. Suitable for Grades 10-12 students.

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