Key Concepts and Formulas for an Ellipse

For an ellipse with its major axis along the x-axis and its center at the origin, its standard equation is given by: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where:

• $a$ is the length of the semi-major axis (half the major axis length).
• $b$ is the length of the semi-minor axis (half the minor axis length).
• For this orientation (major axis along x-axis), we always have $a > b$.

Other important formulas and relationships for this type of ellipse are:

• Eccentricity ($e$): A measure of how "oval" the ellipse is. It is related to $a$ and $b$ by the equation: $b^2 = a^2(1-e^2)$ Alternatively, $e = \frac{c}{a}$, where $c$ is the distance from the center to each focus. From $b^2 = a^2 - a^2e^2$, we get $a^2e^2 = a^2 - b^2$, so $c^2 = a^2 - b^2$, which implies $c = \sqrt{a^2 - b^2}$.
• Length of the Latus Rectum: The length of the chord passing through a focus and perpendicular to the major axis is: $L = \frac{2b^2}{a}$
• Distance between Foci: The foci are located at $(\pm ae, 0)$. The distance between them is: $D_f = 2ae$
• Length of the Minor Axis: The length of the minor axis is $2b$.

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Step-by-Step Derivation

Step 1: Formulate Equations from the Given Information

We are given two pieces of information about the ellipse:

1. The length of the latus rectum is 8.

   • Using the formula for the length of the latus rectum, $L = \frac{2b^2}{a}$, we can write: $\frac{2b^2}{a} = 8$
   • Simplifying this equation, we get our first relationship between $a$ and $b$: $b^2 = 4a \quad \text{(Equation 1)}$
   • Explanation: This step translates the given verbal description into a mathematical equation using the standard formula for an ellipse.

2. The distance between the foci is equal to the length of its minor axis.

   • Using the formula for the distance between foci, $D_f = 2ae$, and the length of the minor axis, $2b$, we can write: $2ae = 2b$
   • Simplifying this equation, we get our second relationship: $ae = b \quad \text{(Equation 2)}$
   • Explanation: Again, we convert the problem statement into a mathematical equation using the relevant definitions. This equation directly relates $a$, $b$, and the eccentricity $e$.

Step 2: Determine the Values of $a$ and $b$

Now we have a system of equations involving $a$, $b$, and $e$. We also know the fundamental relation $b^2 = a^2(1-e^2)$. Our goal is to find $a$ and $b$ to determine the equation of the ellipse.

1. Relate $a$ and $b$ using eccentricity:
   • From Equation 2, $ae = b$, we can express eccentricity as $e = \frac{b}{a}$.
   • Substitute this into the fundamental relation $b^2 = a^2(1-e^2)$: $b^2 = a^2 \left( 1 - \left(\frac{b}{a}\right)^2 \right)$