Focal Parameter of Hyperbola given Eccentricity Calculator

Formula Used:

\[ p = \frac{b^2}{a \times e} \]

Semi Conjugate Axis of Hyperbola (b):

Semi Transverse Axis of Hyperbola (a):

Eccentricity of Hyperbola (e):

Focal Parameter of Hyperbola (p):

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1. What is the Focal Parameter of Hyperbola?

The Focal Parameter of Hyperbola is the shortest distance between any of the foci and directrix of the corresponding wing of the Hyperbola. It is an important geometric property that helps define the shape and orientation of a hyperbola.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ p = \frac{b^2}{a \times e} \]

Where:

\( p \) — Focal Parameter of Hyperbola (m)
\( b \) — Semi Conjugate Axis of Hyperbola (m)
\( a \) — Semi Transverse Axis of Hyperbola (m)
\( e \) — Eccentricity of Hyperbola (m)

Explanation: This formula calculates the focal parameter using the semi-conjugate axis, semi-transverse axis, and eccentricity of the hyperbola.

3. Importance of Focal Parameter Calculation

Details: The focal parameter is crucial in understanding the geometric properties of hyperbolas, particularly in optics, astronomy, and engineering applications where hyperbolic shapes are used.

4. Using the Calculator

Tips: Enter the semi-conjugate axis (b), semi-transverse axis (a), and eccentricity (e) in meters. All values must be positive numbers greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is the relationship between focal parameter and other hyperbola parameters?
A: The focal parameter is directly related to the semi-conjugate axis and inversely related to both the semi-transverse axis and eccentricity.

Q2: Can the focal parameter be zero?
A: No, the focal parameter cannot be zero as it represents a physical distance between the focus and directrix.

Q3: How does eccentricity affect the focal parameter?
A: As eccentricity increases, the focal parameter decreases, assuming other parameters remain constant.

Q4: What are typical applications of this calculation?
A: This calculation is used in optics for hyperbolic mirrors, in astronomy for orbital mechanics, and in various engineering applications involving hyperbolic geometries.

Q5: Are there limitations to this formula?
A: This formula applies specifically to standard hyperbolas and assumes all input parameters are accurately measured and positive.