Semi Latus Rectum Of Hyperbola Given Focal Parameter And Semi Conjugate Axis Calculator

Formula Used:

\[ L = \frac{b \times p}{\sqrt{b^2 - p^2}} \]

Semi Latus Rectum of Hyperbola (L):

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1. What is Semi Latus Rectum of Hyperbola?

The Semi Latus Rectum of a Hyperbola is half the length of the latus rectum, which is the chord through the focus perpendicular to the major axis. It represents an important geometric property of hyperbolas in conic sections.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ L = \frac{b \times p}{\sqrt{b^2 - p^2}} \]

Where:

\( L \) — Semi Latus Rectum of Hyperbola (m)
\( b \) — Semi Conjugate Axis of Hyperbola (m)
\( p \) — Focal Parameter of Hyperbola (m)

Explanation: This formula calculates the semi latus rectum using the semi conjugate axis and focal parameter of the hyperbola, derived from the geometric properties of conic sections.

3. Importance of Semi Latus Rectum Calculation

Details: The semi latus rectum is crucial for understanding the geometric properties of hyperbolas, including their focal points, directrices, and overall shape. It's particularly important in orbital mechanics and optical systems.

4. Using the Calculator

Tips: Enter the semi conjugate axis (b) and focal parameter (p) in meters. Both values must be positive, and b must be greater than p for valid calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between semi latus rectum and other hyperbola parameters?
A: The semi latus rectum relates to both the semi transverse axis and eccentricity, providing insight into the hyperbola's shape and focal properties.

Q2: How does semi latus rectum differ from latus rectum?
A: Semi latus rectum is exactly half the length of the latus rectum, which is the full chord through the focus perpendicular to the major axis.

Q3: What are typical applications of semi latus rectum calculations?
A: Used in astronomy for orbital calculations, in physics for particle trajectories, and in engineering for designing reflective surfaces.

Q4: Are there limitations to this formula?
A: The formula assumes a standard hyperbola and may not apply to degenerate cases or hyperbolas with special orientations.

Q5: How does semi latus rectum relate to hyperbola's eccentricity?
A: For hyperbolas, the semi latus rectum can be expressed as \( L = a(e^2 - 1) \), where a is the semi transverse axis and e is the eccentricity.