This solution will guide you through the problem step-by-step, explaining the underlying concepts and formulas for hyperbolas.

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1. Understanding the Hyperbola and its Standard Form

The given equation of the hyperbola is $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$. The first crucial step in any conic section problem is to identify its standard form and orientation. We can rearrange the terms to get: $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ Key Concept: This is the standard form of a hyperbola where the transverse axis lies along the y-axis. This means the major features are oriented vertically:

• The vertices are at $(0, \pm b)$.
• The foci are at $(0, \pm be)$.
• The directrices are $y = \pm \frac{b}{e}$.
• The relationship between the semi-conjugate axis ($a$), semi-transverse axis ($b$), and eccentricity ($e$) is given by $a^2 = b^2(e^2-1)$.
• The length of the latus rectum ($LR$) is $\frac{2a^2}{b}$.

Common Mistake: Do not confuse this with a hyperbola whose transverse axis is along the x-axis ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$). For that type, the roles of $a$ and $b$ are swapped in the eccentricity relation ($b^2 = a^2(e^2-1)$) and latus rectum formula ($LR = \frac{2b^2}{a}$), and the foci/vertices are on the x-axis. Always check which term is positive in the standard equation to determine the transverse axis.

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2. Determining the Hyperbola Parameters 'a' and 'b'

We are given two properties of the hyperbola:

1. Eccentricity, $e = \sqrt{3}$.
2. Length of the latus rectum, $LR = 4\sqrt{3}$.

We will use the formulas specific to a hyperbola with its transverse axis along the y-axis.

Step 2a: Using the Eccentricity to relate 'a' and 'b' The relationship for eccentricity is $a^2 = b^2(e^2-1)$. Substitute the given value $e = \sqrt{3}$: $a^2 = b^2((\sqrt{3})^2 - 1)$ $a^2 = b^2(3 - 1)$ $a^2 = 2b^2 \quad \text{(Equation 1)}$ Explanation: This equation links the semi-conjugate axis ($a$) and semi-transverse axis ($b$) through the eccentricity. It's a fundamental property that defines the "shape" or "openness" of the hyperbola.

Step 2b: Using the Length of the Latus Rectum to relate 'a' and 'b' The formula for the length of the latus rectum is $LR = \frac{2a^2}{b}$. Substitute the given value $LR = 4\sqrt{3}$: $4\sqrt{3} = \frac{2a^2}{b}$ Divide both sides by 2: $2\sqrt{3} = \frac{a^2}{b}$ $a^2 = 2\sqrt{3}b \quad \text{(Equation 2)}$ Explanation: The latus rectum is a chord passing through a focus and perpendicular to the transverse axis. Its length provides another geometric constraint on the dimensions of the hyperbola.

Step 2c: Solving for 'a' and 'b' Now we have a system of two equations with two unknowns ($a^2$ and $b$):

1. $a^2 = 2b^2$
2. $a^2 = 2\sqrt{3}b$

Equating the expressions for $a^2$ from both equations: $2b^2 = 2\sqrt{3}b$ Since $b$ represents a length