Write the standard form of the equation of the circle with the given characteristics. Center: (6, −4) ; Radius: $$ \sqrt{19} $$

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 and $r$ is the radius.

Given:
- Center: $(6, -4)$
- Radius: $ \sqrt{19} $

Substituting the values into the standard form:

\[
(x - 6)^2 + (y + 4)^2 = (\sqrt{19})^2
\]

Simplifying:

\[
(x - 6)^2 + (y + 4)^2 = 19
\]

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

\[
(x - 6)^2 + (y + 4)^2 = 19
\]

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

Here are five related questions you might find helpful:
1. What is the general form of a circle's equation?
2. How do you convert the general form of a circle equation to standard form?
3. How do you find the center and radius from an equation of a circle?
4. What happens to the equation of a circle if you translate the center?
5. How can you derive the equation of a circle from its diameter?

**Tip:** Always square the radius when writing the circle equation in standard form.

Write the standard form of the equation of the circle with the given characteristics. Center: (6, −4) ; Radius: $$ \sqrt{19} $$

Learn how to write the standard form of the equation of a circle with center (6, -4) and radius sqrt(19). This problem covers essential geometry concepts and the formula for the standard equation of a circle. Suitable for students in Grades 9-11.

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