Linear Eccentricity of Hyperbola Given Latus Rectum and Semi Conjugate Axis Calculator

Formula Used:

\[ c = \sqrt{\frac{b^2}{1 - \frac{1}{1 + \frac{L^2}{(2b)^2}}}} \]

Linear Eccentricity of Hyperbola (c):

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1. What is Linear Eccentricity of Hyperbola?

Linear Eccentricity of a hyperbola is half of the distance between the two foci of the hyperbola. It is an important parameter that helps define the shape and properties of the hyperbola.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ c = \sqrt{\frac{b^2}{1 - \frac{1}{1 + \frac{L^2}{(2b)^2}}}} \]

Where:

\( c \) — Linear Eccentricity of Hyperbola (m)
\( b \) — Semi Conjugate Axis of Hyperbola (m)
\( L \) — Latus Rectum of Hyperbola (m)

Explanation: This formula calculates the linear eccentricity based on the semi-conjugate axis and latus rectum of the hyperbola, using square root and algebraic operations.

3. Importance of Linear Eccentricity Calculation

Details: Calculating linear eccentricity is crucial for understanding the geometric properties of hyperbolas, including their foci positions, shape characteristics, and relationships between different parameters in conic section mathematics.

4. Using the Calculator

Tips: Enter the semi-conjugate axis (b) and latus rectum (L) values in meters. Both values must be positive numbers greater than zero for accurate calculation.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between linear eccentricity and other hyperbola parameters?
A: Linear eccentricity is related to both the transverse and conjugate axes through the formula \( c^2 = a^2 + b^2 \), where a is the semi-transverse axis.

Q2: Can linear eccentricity be larger than the semi-transverse axis?
A: Yes, for hyperbolas, linear eccentricity is always greater than the semi-transverse axis (c > a).

Q3: What are typical units for these measurements?
A: While meters are used here, any consistent unit of length can be used (cm, mm, etc.) as long as all inputs use the same unit.

Q4: How does latus rectum relate to the hyperbola's shape?
A: The latus rectum is a chord through the focus perpendicular to the transverse axis, and its length helps determine the hyperbola's eccentricity and shape.

Q5: Are there limitations to this formula?
A: This specific formula applies when you have the semi-conjugate axis and latus rectum values. Different formulas are used when other parameters are known.