In problems, 13 through 22 write the standard form of the equation in the general form of the equation of each circle of radius r And Center (h,k) Graph each circle 15) R= 2 (h,k) =(0,2)

To solve this problem, we need to write the equation of a circle in standard form and convert it to general form. The standard form of the equation of a circle with radius \( r \) and center \( (h, k) \) is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Given:
- Radius \( r = 2 \)
- Center \( (h, k) = (0, 2) \)

### Step 1: Write the equation in standard form

Substitute the values of \( h \), \( k \), and \( r \) into the standard form equation:

\[
(x - 0)^2 + (y - 2)^2 = 2^2
\]

Simplify:

\[
x^2 + (y - 2)^2 = 4
\]

### Step 2: Expand to convert to general form

Now, expand \( (y - 2)^2 \):

\[
x^2 + (y^2 - 4y + 4) = 4
\]

Combine like terms:

\[
x^2 + y^2 - 4y + 4 = 4
\]

Subtract 4 from both sides to set the equation to 0:

\[
x^2 + y^2 - 4y = 0
\]

### Conclusion

The general form of the equation of the circle is:

\[
x^2 + y^2 - 4y = 0
\]

### Graphing the Circle

To graph the circle:
1. Plot the center at \( (0, 2) \).
2. Since the radius is 2, draw a circle with a radius of 2 units centered at \( (0, 2) \).

Would you like more details or have any questions about this solution?

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Here are 5 related questions to further your understanding:

1. How do you find the radius and center of a circle given its equation in general form?
2. What is the process to convert from general form to standard form for a circle equation?
3. How do you complete the square to convert a general form equation to the standard form of a circle?
4. What is the effect on the circle's graph if the radius is doubled?
5. How does the position of the center affect the graph of the circle?

**Tip:** Remember, when graphing a circle, always ensure the radius is drawn accurately from the center to maintain the circle's correct size and shape.

Learn how to convert the equation of a circle with radius 2 and center at (0, 2) from standard form to general form. This problem involves applying the equation of a circle formula, standard form conversion, and graphical representation. Suitable for Grades 9-12.

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