The standard form of an ellipse can be written as:

$\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$

where:

• $(h, k)$ is the center of the ellipse,
• $a$ is the semi-major axis (half the length of the major axis),
• $b$ is the semi-minor axis (half the length of the minor axis),
• $c$ is the distance from the center to the foci.

Step 1: Find the center $(h, k)$

The foci of the ellipse are given as $(1, 5)$ and $(1, 1)$. Since the foci lie on a vertical line, the major axis is vertical. The center of the ellipse is the midpoint of the two foci:

$h = \frac{1 + 1}{2} = 1, \quad k = \frac{5 + 1}{2} = 3$

Thus, the center of the ellipse is $(1, 3)$.

Step 2: Determine $c$

The distance between the center and one of the foci is $c$, which is half the distance between the foci. The distance between the foci is:

$|5 - 1| = 4$

Thus:

$c = \frac{4}{2} = 2$

Step 3: Find $b$

The length of the minor axis is given as 4, so the semi-minor axis $b$ is:

$b = \frac{4}{2} = 2$

Step 4: Determine $a$

The relationship between $a$, $b$, and $c$ in an ellipse is given by:

$a^2 = b^2 + c^2$

Substituting the values of $b = 2$ and $c = 2$:

$a^2 = 2^2 + 2^2 = 4 + 4 = 8$

Thus:

$a = \sqrt{8} = 2\sqrt{2}$

Step 5: Write the equation of the ellipse

Now that we have the values of $a$, $b$, and the center $(h, k)$, the equation of the ellipse is:

$\frac{(x - 1)^2}{2^2} + \frac{(y - 3)^2}{(2\sqrt{2})^2} = 1$

Simplifying:

$\frac{(x - 1)^2}{4} + \frac{(y - 3)^2}{8} = 1$

This is the standard form of the ellipse.

Would you like further clarification or additional details?

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Here are 5 related questions to explore:

1. How do you find the lengths of the axes for a rotated ellipse?
2. What is the relationship between eccentricity and the foci of an ellipse?
3. How does the formula for an ellipse change when it is not centered at the origin?
4. Can you derive the formula for the area of an ellipse?
5. How would the equation change if the ellipse was horizontal?

Tip: In an ellipse, the major axis is always longer than the minor axis unless the ellipse is a circle.