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Step-by-Step Solution
Step 1: Recognize the Orientation of the Ellipse
The ellipse is centered at the origin and has one of its foci on the y-axis at $(0,5\sqrt{3})$. This indicates the major axis is along the y-axis.
Step 2: Interpret the Given Information
1. One focus is at $(0,\,be)$, where $b$ is the semi-major axis and $a$ is the semi-minor axis (because $b > a$ for a vertically oriented ellipse). Given that focus is $(0,5\sqrt{3})$, we have:
$be = 5\sqrt{3}$.
2. The difference of the lengths of the major axis and minor axis is 10. Hence,
$2b - 2a = 10 \quad \Rightarrow \quad b - a = 5.$
Step 3: Relate Eccentricity to $a$ and $b$
For an ellipse with vertical major axis,
$e^2 = 1 - \frac{a^2}{b^2}, \quad \text{where } e \text{ is the eccentricity.}$
Since $be = 5\sqrt{3}$,
$b \cdot e = 5\sqrt{3} \quad \Rightarrow \quad b^2 e^2 = (5\sqrt{3})^2 = 75.
Using $e^2 = 1 - \frac{a^2}{b^2}$, we get:
$b^2 \Bigl(1 - \frac{a^2}{b^2}\Bigr) = 75 \quad \Rightarrow \quad b^2 - a^2 = 75. \quad (1)
Step 4: Use the System of Equations to Find $a$ and $b$
From the difference of semi-axis lengths, we also have:
$b - a = 5. \quad (2)
Equation (1) can be factored as:
$(b + a)(b - a) = 75.
Substituting from (2), $(b - a) = 5$, we get:
$(b + a)\cdot 5 = 75 \quad \Rightarrow \quad b + a = 15.
Solving the system:
$\begin{cases}
b - a = 5 \\
b + a = 15
\end{cases}$
Adding the two equations: $2b = 20 \Rightarrow b = 10.$
Subtracting gives: $2a = 10 \Rightarrow a = 5.$
Step 5: Compute the Length of the Latus Rectum
For an ellipse (vertically oriented) with major axis along $y$, the length of the latus rectum is given by:
$\text{Latus Rectum} = \frac{2a^2}{b}.$
Substitute $a = 5$ and $b = 10$:
$\frac{2\times (5)^2}{10} = \frac{2 \times 25}{10} = 5.$
Thus, the length of the latus rectum is $\boxed{5}.$
