Semi Latus Rectum of Hyperbola given Eccentricity and Semi Transverse Axis Calculator

Semi Latus Rectum of Hyperbola Formula:

\[ L_{Semi} = a \times (e^2 - 1) \]

Semi Latus Rectum of Hyperbola (L_Semi):

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1. What is the Semi Latus Rectum of Hyperbola?

The Semi Latus Rectum of Hyperbola is half of the line segment passing through any of the foci and perpendicular to the transverse axis whose ends are on the Hyperbola. It is an important parameter in describing the geometric properties of hyperbolas.

2. How Does the Calculator Work?

The calculator uses the Semi Latus Rectum formula:

\[ L_{Semi} = a \times (e^2 - 1) \]

Where:

\( L_{Semi} \) — Semi Latus Rectum of Hyperbola (m)
\( a \) — Semi Transverse Axis of Hyperbola (m)
\( e \) — Eccentricity of Hyperbola

Explanation: The formula calculates the semi latus rectum based on the semi transverse axis and eccentricity of the hyperbola, which defines its shape and size.

3. Importance of Semi Latus Rectum Calculation

Details: Calculating the semi latus rectum is crucial for understanding the geometric properties of hyperbolas, including focal distances, directrices, and the overall shape of the conic section in mathematical and engineering applications.

4. Using the Calculator

Tips: Enter the semi transverse axis in meters and the eccentricity value. Both values must be positive numbers (a > 0, e > 0) for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between semi latus rectum and eccentricity?
A: The semi latus rectum increases with both the semi transverse axis and the square of eccentricity minus one, as shown in the formula L_Semi = a × (e² - 1).

Q2: Can semi latus rectum be negative?
A: No, since both semi transverse axis and eccentricity are positive values, and e² is always ≥ 1 for hyperbolas, the semi latus rectum is always positive.

Q3: How does semi latus rectum relate to the focal parameter?
A: The semi latus rectum is exactly half of the focal parameter (latus rectum) of the hyperbola.

Q4: What are typical values for hyperbola eccentricity?
A: For hyperbolas, eccentricity is always greater than 1 (e > 1), distinguishing them from ellipses (e < 1) and parabolas (e = 1).

Q5: How is this calculation used in real-world applications?
A: Hyperbola geometry calculations are used in various fields including astronomy (orbital mechanics), physics (particle trajectories), engineering (reflector design), and navigation systems.