Semi Transverse Axis of Hyperbola given Eccentricity and Focal Parameter Calculator

Formula Used:

\[ a = \frac{p \times e}{e^2 - 1} \]

Semi Transverse Axis of Hyperbola (a):

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1. What is the Semi Transverse Axis of Hyperbola?

The Semi Transverse Axis of Hyperbola is half of the distance between the vertices of the Hyperbola. It is a fundamental parameter that defines the size and shape of the hyperbola.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ a = \frac{p \times e}{e^2 - 1} \]

Where:

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

Explanation: This formula calculates the semi transverse axis using the focal parameter and eccentricity of the hyperbola, which are related through the geometric properties of conic sections.

3. Importance of Semi Transverse Axis Calculation

Details: The semi transverse axis is crucial for determining the overall dimensions of the hyperbola, its asymptotes, and other geometric properties. It's essential in various applications including optics, astronomy, and engineering design.

4. Using the Calculator

Tips: Enter the focal parameter and eccentricity values in meters. Both values must be positive numbers, and eccentricity must not equal 1 (which would make the denominator zero).

5. Frequently Asked Questions (FAQ)

Q1: What is the range of valid values for eccentricity?
A: For hyperbolas, eccentricity must be greater than 1. Values less than or equal to 1 correspond to other conic sections (ellipse or parabola).

Q2: Can eccentricity be exactly 1?
A: No, eccentricity cannot be exactly 1 as this would make the denominator zero, resulting in an undefined value for the semi transverse axis.

Q3: What are typical units for these measurements?
A: While meters are used here, any consistent unit system can be applied as long as all inputs use the same units.

Q4: How does the focal parameter relate to other hyperbola parameters?
A: The focal parameter is related to the semi transverse axis and eccentricity through the formula \( p = a(e^2 - 1)/e \).

Q5: What if I get a negative result?
A: The semi transverse axis should always be positive. A negative result indicates invalid input values or calculation error.