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The Semi Conjugate Axis of Hyperbola is half of the tangent from any of the vertices of the Hyperbola and chord to the circle passing through the foci and centered at the center of the Hyperbola. It represents the distance from the center to the co-vertices along the conjugate axis.
The calculator uses the formula:
Where:
Explanation: This formula derives from the fundamental relationship between the semi-axes and linear eccentricity in a hyperbola, where the linear eccentricity is always greater than the semi-transverse axis.
Details: The formula uses the square root function to calculate the semi conjugate axis based on the Pythagorean relationship between the three parameters. The linear eccentricity (c) represents the distance from the center to either focus, while a and b are the semi-axes.
Tips: Enter linear eccentricity and semi transverse axis values in meters. Both values must be positive, and the linear eccentricity must be greater than the semi transverse axis for a valid result.
Q1: What are the units of measurement?
A: The calculator uses meters (m) for all measurements, but the formula works with any consistent unit system.
Q2: Why must linear eccentricity be greater than semi transverse axis?
A: In a hyperbola, the distance to the foci (linear eccentricity) is always greater than the distance to the vertices (semi transverse axis) by definition.
Q3: What if I get an imaginary result?
A: An imaginary result indicates that the input values don't represent a valid hyperbola. Ensure that c > a.
Q4: Can this formula be used for ellipses?
A: No, for ellipses the relationship is different: \( c = \sqrt{a^2 - b^2} \) where a is the semi-major axis.
Q5: What is the geometric significance of the semi conjugate axis?
A: The semi conjugate axis determines the "spread" of the hyperbola and affects the slopes of the asymptotes.