C (-4,6) r:5 what is the standard form to general form

The equation of a circle in **standard form** is:

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

Where:
- \( C(h, k) \) is the center of the circle.
- \( r \) is the radius.

Given:
- \( C(-4, 6) \) is the center, so \( h = -4 \) and \( k = 6 \).
- \( r = 5 \), so \( r^2 = 25 \).

Substituting these values into the standard form:

\[
(x - (-4))^2 + (y - 6)^2 = 5^2
\]

Simplify:

\[
(x + 4)^2 + (y - 6)^2 = 25
\]

This is the **standard form** of the circle.

### To convert this to **general form**, expand the equation:

\[
(x + 4)^2 + (y - 6)^2 = 25
\]

Expand both squared terms:

\[
(x^2 + 8x + 16) + (y^2 - 12y + 36) = 25
\]

Combine like terms:

\[
x^2 + y^2 + 8x - 12y + 52 = 25
\]

Finally, subtract 25 from both sides:

\[
x^2 + y^2 + 8x - 12y + 27 = 0
\]

This is the **general form** of the circle.

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Would you like any further details or explanations? Here are some related questions you might be interested in:

1. How can we verify the general form of a circle equation?
2. What is the difference between the general form and the standard form?
3. How do you find the center and radius from the general form?
4. How can we complete the square to convert general form back to standard form?
5. What are some common applications of circle equations?

**Tip:** Always check that your final equation satisfies the original center and radius by substituting back.

Learn how to convert the equation of a circle from standard form to general form with a step-by-step example using the center (-4, 6) and radius 5. This guide covers key geometry concepts for students in Grades 9-11.

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