In a certain ellipse, the center is at $(-3,1),$ one focus is at $(-3,0),$ and one endpoint of a semi-major axis is at $(-3,3).$  Find the semi-minor axis of the ellipse.
The distance between the center and the focus $(-3,0)$ is $c = 1.$  Also, the semi-major axis is the distance between the center and the endpoint of the semi-major axis, which is $a = 2.$  Then the semi-minor axis is $b = \sqrt{a^2 - c^2} = \boxed{\sqrt{3}}.$
\sqrt{3}
Level 4
Intermediate Algebra
Problem: In a certain ellipse, the center is at $(-3,1),$ one focus is at $(-3,0),$ and one endpoint of a semi-major axis is at $(-3,3).$  Find the semi-minor axis of the ellipse.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the semi-minor axis of the ellipse, we first identify the key components of the ellipse based on the given information. The center of the ellipse is at the point (-3, 1). The distance from the center to the endpoint of the semi-major axis at (-3, 3) indicates that the semi-major axis length is 2 units, as it is the vertical distance from the center (1) to the endpoint (3).

Next, we determine the distance from the center to the focus at (-3, 0). This distance is 1 unit, which represents the value of \(c\), where \(c\) is the distance from the center to each focus of the ellipse.

The relationship between the semi-major axis \(a\), the semi-minor axis \(b\), and the distance to the foci \(c\) in an ellipse is given by the equation:

\[ c^2 = a^2 - b^2 \]

We already established that \(a = 2\) and \(c = 1\). We can now substitute these values into the equation:

1. Calculate \(a^2\):
   \[ a^2 = 2^2 = 4 \]

2. Calculate \(c^2\):
   \[ c^2 = 1^2 = 1 \]

3. Substitute into the equation:
   \[ 1 = 4 - b^2 \]

4. Rearranging gives:
   \[ b^2 = 4 - 1 = 3 \]

5. Taking the square root to find \(b\):
   \[ b = \sqrt{3} \]

Thus, the semi-minor axis of the ellipse is \(\sqrt{3}\).<|END_OF_LATENT|> The final answer is $\sqrt{3}$.