Show that the length of conjugate axis of a hyperbola is the mean proportional between the length of its transverse axis and the length of its latus rectum.

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Solution

Length of transverse axis is $2 a$
Length of conjugate axis is $2 b$
Length of latus rectum is $2 a (e^{2} - 1)$
mean of length of latus rectum and transverse axis is $\frac{2 a + 2 a (e^{2} - 1)}{2} = a e^{2} = \frac{e^{2}}{2 \sqrt{e^{2} - 1}} (2 a \sqrt{e^{2} - 1}) = \frac{e^{2}}{2 \sqrt{e^{2} - 1}} (2 b)$
I.e. Length of conjugate axis of a hyperbola is the mean proportional between the length of transverse axis and the length of its latus rectum

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