what is the standard and general equation?

### Standard Equation of a Circle

The **standard equation** of a circle with center at \((h, k)\) and radius \(r\) is:

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

Here:
- \((h, k)\) represents the coordinates of the center of the circle.
- \(r\) is the radius of the circle.

### General Equation of a Circle

The **general equation** of a circle can be derived from the standard form by expanding and rearranging the terms. It is usually written as:

\[
Ax^2 + Ay^2 + Dx + Ey + F = 0
\]

For a circle, \(A\) is a non-zero constant, and \(A\) is the same for both \(x^2\) and \(y^2\) terms (since a circle is symmetric).

#### Derivation from Standard to General Form:

Starting from the standard equation:

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

Expand the squares:

\[
x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2
\]

Now, rearrange to form the general equation:

\[
x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0
\]

This can be written as:

\[
x^2 + y^2 + Dx + Ey + F = 0
\]

Where:
- \(D = -2h\)
- \(E = -2k\)
- \(F = h^2 + k^2 - r^2\)

### Example with \(h = -5\), \(k = -\frac{1}{2}\), \(r = \frac{1}{2}\):

Given the center \( (-5, -\frac{1}{2}) \) and radius \( \frac{1}{2} \), the standard equation was derived as:

\[
(x + 5)^2 + \left( y + \frac{1}{2} \right)^2 = \frac{1}{4}
\]

Expanding this to the general form:

1. Expand the squares:

\[
(x^2 + 10x + 25) + \left( y^2 + y + \frac{1}{4} \right) = \frac{1}{4}
\]

2. Combine and simplify:

\[
x^2 + y^2 + 10x + y + 25 + \frac{1}{4} = \frac{1}{4}
\]

3. Subtract \(\frac{1}{4}\) from both sides:

\[
x^2 + y^2 + 10x + y + 25 = 0
\]

This is the general equation of the circle.

Would you like more details on this, or do you have any further questions?

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Here are some related questions to further your understanding:
1. How do you convert from the general form back to the standard form of a circle?
2. What does each term in the general equation of a circle represent geometrically?
3. Can a circle have a general equation where \(A\) is different for \(x^2\) and \(y^2\)?
4. What happens to the general equation if the circle passes through the origin?
5. How do you determine the center and radius of a circle from its general equation?
6. How does completing the square help in deriving the standard form from the general form?
7. What conditions on \(D\), \(E\), and \(F\) must be met for a valid circle equation?
8. How would you graph a circle if given the general equation directly?

**Tip:** In the general equation of a circle, if the coefficients of \(x^2\) and \(y^2\) are equal but not 1, you can factor them out to convert it to the standard form.

Explore the standard and general equations of a circle with detailed explanations and examples. Learn how to convert between the standard form \((x - h)^2 + (y - k)^2 = r^2\) and the general form \(Ax^2 + Ay^2 + Dx + Ey + F = 0\). Suitable for high school students and anyone interested in geometry and circle equations.

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