Relation between focal distances:
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The difference of focal distances of a point on hyperbola is constant. PM and PM' are perpendiculars to directrices MZ and M'Z'
PS' - PS = e(PM' - PM)
= eMM' = e(2a/e) = 2a = constant.
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Relative position of a point with respect to the hyperbola
The quantity 
is positive, zero or negative, as the point (x1, y1) lies within, upon or without the curve.
Parametric coordinates:
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Thus Any point on curve, in the parametric form is
X = a secθ, y = b tanθ.
Or we can say that (a secθ, b tanθ) is a point on hyperbola for all values of q. The point (a secθ, b tanθ) is briefly called as point 'θ'.
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