Conjugate hyperbola:

The hyperbola, whose transverse and conjugate axes are the conjugate and transverse axes of a given hyperbola respectively, is called as conjugate hyperbola of the given hyperbola, and the 2 hyperbolas are conjugate to one another. Therefore, the hyperbolas  and  are the conjugate hyperbolas.

Illustration: Tangents are drawn to the hyperbola from any point on one of the branches of conjugate hyperbola. Show that their chord of contact will touch other branch of conjugate hyperbola.

Solution: Let the hyperbola be  = 1. So the conjugate hyperbola is  = - 1. Let any point on it be (a tanθ, b secθ). Now the equation of chord of contact will be

So, it is a tangent to conjugate hyperbola at the point (- a tanθ, - b secθ) which will obviously on other branch. 

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