Semi Transverse Axis of Hyperbola given Linear Eccentricity and Focal Parameter Calculator

Formula Used:

\[ a = \sqrt{c^2 - (p \times c)} \]

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 represents the distance from the center to either vertex along the transverse axis.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ a = \sqrt{c^2 - (p \times c)} \]

Where:

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

Explanation: This formula calculates the semi transverse axis using the linear eccentricity and focal parameter of the hyperbola, derived from the geometric properties of conic sections.

3. Importance of Semi Transverse Axis Calculation

Details: The semi transverse axis is a fundamental parameter in hyperbola geometry, essential for determining the shape, size, and various other properties of the hyperbola in mathematical and engineering applications.

4. Using the Calculator

Tips: Enter linear eccentricity (c) and focal parameter (p) 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 semi transverse axis and linear eccentricity?
A: The semi transverse axis (a) and linear eccentricity (c) are related through the formula \( c^2 = a^2 + b^2 \), where b is the semi conjugate axis.

Q2: Can the semi transverse axis be larger than the linear eccentricity?
A: No, in a hyperbola, the linear eccentricity (c) is always greater than the semi transverse axis (a).

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

Q4: What if I get a negative value under the square root?
A: This indicates invalid input values. For a hyperbola, the relationship \( c > a \) must hold, and the focal parameter must be appropriately related to c.

Q5: How is this formula derived?
A: The formula is derived from the fundamental relationship between the linear eccentricity, focal parameter, and semi transverse axis in hyperbola geometry.