Let's start by analyzing the equation of the circle given in its standard form:

$x^2 + (y - 1)^2 = 1$

The general form of a circle's equation is:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

• $(h, k)$ is the center of the circle.
• $r$ is the radius.

Step-by-step solution:

1. Compare the given equation with the standard form: $x^2 + (y - 1)^2 = 1$

   • The term $(x - h)^2$ is just $x^2$, which means $h = 0$.
   • The term $(y - k)^2$ is $(y - 1)^2$, so $k = 1$.
   • The right side of the equation is $r^2 = 1$, meaning the radius is $r = 1$.

2. Center and radius:

   • The center of the circle is $(0, 1)$.
   • The radius of the circle is $1$.

Thus:

• Center: $(0, 1)$
• Radius: $1$

Would you like more details on how to graph the circle, or do you have any questions?

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Related questions to explore:

1. How do you graph a circle given its equation in standard form?
2. How does the equation change if the center of the circle is shifted to another point?
3. What happens to the circle's equation if the radius is doubled?
4. How would you rewrite the equation if the center is $(-2, 3)$ and the radius is 4?
5. How can you derive the equation of a circle given the endpoints of its diameter?

Tip:

Remember, the general equation of a circle provides the center and radius directly. Just compare terms to identify these values!