how is it that the b squared in the equation of the hyperbola is equal to the sqaure of the semi conjugate axis with length b?

Question

what is the significance of the b in value of the conjugate axis?

Answer 1

During the DEVELOPMENT of the EQUATION of an hyperbola, such as the one that eventually becomes x²/a² - y²/b² = 1, the beginning conditions are the adopting of the transverse axis coinciding with the x-axis, and the 'center' at the origin. This includes placing the foci at (-c,o) & (c,0) on the transverse axis.
Closer to the origin, points called vertices are placed at (-a,0) & (a,0).
Then in the course of the algebra that is developed based on the difference in distances from P(x,y) on the hyperbola to the foci, the steps lead to frequent appearance of the expression (c² - a²). Obviously this difference is always positive.
Therefore the artificial expression b² is adopted to replace this...eventually leading to the standard equation stated above.

This is the PROOF that b² = (c² - a²), consistent with what you requested.

"b" is not even part of the original conditions under which the hyperbola's definition is stated. It does turn out, as you may know, that "b" has a real interpretation, in connection with asymptotes.

This is a cut & paste of an answer to a similar questions I answered yesterday