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

Formula Used:

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

Semi Latus Rectum (L_Semi):

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1. What is 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 geometry of a hyperbola.

2. How Does the Calculator Work?

The calculator uses the formula:

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

Where:

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

Explanation: The formula calculates the semi latus rectum based on the semi conjugate axis and eccentricity of the hyperbola, which are fundamental parameters defining its shape.

3. Importance of Semi Latus Rectum Calculation

Details: The semi latus rectum is crucial in conic section geometry as it relates to the focal parameter and helps in determining various properties of the hyperbola, including its curvature and the relationship between different axes.

4. Using the Calculator

Tips: Enter the semi conjugate axis in meters (must be positive) and eccentricity (must be greater than 1). The calculator will compute the semi latus rectum of the hyperbola.

5. Frequently Asked Questions (FAQ)

Q1: What is the physical significance of semi latus rectum?
A: The semi latus rectum represents half the length of the chord through a focus perpendicular to the major axis, providing important geometric information about the hyperbola's shape.

Q2: How does eccentricity affect the semi latus rectum?
A: For a fixed semi conjugate axis, as eccentricity increases, the semi latus rectum also increases, indicating a more "stretched" hyperbola.

Q3: Can the semi latus rectum be zero?
A: No, since both semi conjugate axis and eccentricity must be positive with eccentricity > 1, the semi latus rectum will always be positive.

Q4: What are typical values for semi conjugate axis and eccentricity?
A: Semi conjugate axis can vary widely depending on the specific hyperbola, while eccentricity for hyperbolas is always greater than 1, typically ranging from just above 1 to much larger values.

Q5: How is this related to the transverse axis?
A: The semi latus rectum can also be expressed in terms of the semi transverse axis (a) and eccentricity (e) as L = a(e² - 1), showing the relationship between these parameters.