Question 1: The Radius of the Circle

The equation of a circle centered at the origin $(0, 0)$ with a radius $r$ is given by:

$x^2 + y^2 = r^2$

For this specific problem, the radius of the circle is $r = 3$. So, the equation of the circle is:

$x^2 + y^2 = 3^2 = 9$

Question 2: The Exact Distance from the Center to the Point $(2, \sqrt{5})$

To find the distance from the center of the circle $(0, 0)$ to the point $(2, \sqrt{5})$, we can use the distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In this case, the center is $(x_1, y_1) = (0, 0)$ and the point is $(x_2, y_2) = (2, \sqrt{5})$. Substituting into the distance formula:

$d = \sqrt{(2 - 0)^2 + (\sqrt{5} - 0)^2} = \sqrt{2^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3$

Thus, the exact distance from the center $(0, 0)$ to the point $(2, \sqrt{5})$ is 3 units.

Question 3: Does the Point $(2, \sqrt{5})$ Lie on the Circle?

Since the distance from the center to the point $(2, \sqrt{5})$ is exactly 3 units, and the radius of the circle is also 3, the point lies on the circle.

Thus, the point does lie on the circle.

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Summary of answers:

1. The radius of the circle is 3.
2. The exact distance from the center $(0, 0)$ to the point $(2, \sqrt{5})$ is 3 units.
3. The point $(2, \sqrt{5})$ does lie on the circle.

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

Here are some related questions you might find useful:

1. How do you calculate the distance between two points in 3D space?
2. What is the equation of a circle with its center at any point $(h, k)$ and radius $r$?
3. How can you determine if a point lies inside or outside a given circle?
4. How do you derive the distance formula from the Pythagorean theorem?
5. What are the differences between the equations of a circle and an ellipse?

Tip: To check if a point lies on a circle, simply compare the square of the distance from the center to the point with the square of the radius.