Write the standard form of the equation of the circle with the given characteristics. Center: (6,-4) and Radius: $$ \sqrt{17} $$

The standard form of the equation of a circle is:

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

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

Given:
- Center: $(6, -4)$
- Radius: $\sqrt{17}$

Substituting the values into the standard form:

\[
(x - 6)^2 + (y + 4)^2 = (\sqrt{17})^2
\]

Since $(\sqrt{17})^2 = 17$, the equation simplifies to:

\[
(x - 6)^2 + (y + 4)^2 = 17
\]

So, the standard form of the equation of the circle is:

\[
(x - 6)^2 + (y + 4)^2 = 17
\]

Would you like details or have any questions?

Here are 5 related questions:

1. How would the equation change if the center were shifted to $(-6, 4)$?
2. What is the general form of the circle's equation derived from the standard form?
3. How does the equation of a circle change if the radius is doubled?
4. Can the center of a circle be at the origin, and how would the equation look?
5. How do you determine the area of a circle given the radius?

**Tip:** When dealing with the square root of a number as the radius, the standard form equation always simplifies to a simpler constant for $r^2$.

Learn how to write the standard form of the equation of a circle with center (6, -4) and radius √17. This problem covers key geometric concepts, including the standard form of circle equations and simplification of radical expressions. Suitable for high school students.

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