1. Understanding the Fundamental Concepts of an Ellipse

To solve this problem, we need to recall the standard definitions and the crucial relationship for an ellipse. For an ellipse centered at the origin, its standard equation is typically given by $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (assuming the major axis is along the x-axis). The key parameters are:

• $a$: The length of the semi-major axis. This is half the length of the longest diameter of the ellipse.
• $b$: The length of the semi-minor axis. This is half the length of the shortest diameter.
• $e$: The eccentricity of the ellipse, which is a measure of how "stretched out" it is. For an ellipse, $0 < e < 1$.

The fundamental relationship connecting these parameters is: $b^2 = a^2(1-e^2)$ This equation is crucial as it links the semi-major axis, semi-minor axis, and eccentricity. It can also be written as $e^2 = 1 - \frac{b^2}{a^2}$.

We also need the definitions for the lengths mentioned in the problem:

• Length of the minor axis: This is $2b$.
• Distance between the foci: The foci of an ellipse are located at $(\pm ae, 0)$ if the major axis is along the x-axis. Therefore, the distance between the two foci is $2ae$.

2. Setting Up the Given Condition

The problem states: "the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci". Let's translate this statement into a mathematical equation using the definitions from Step 1.

• "Length of the minor axis" is $2b$.
• "Distance between the foci" is $2ae$.
• "One fourth of" means multiplying by $\frac{1}{4}$.

So, the given condition can be written as: $2b = \frac{1}{4} (2ae)$

3. Simplifying the Equation

Now, we will simplify the equation derived