P is a point on the hyperbola x2a2−y2b2=1S S and S′ are its foci. Statement-1: Product of the lengths of the perpendiculars from S and S′ on the tangent at P is equal to Statement-2: PS−PS′=2a.

Let  P(asec⁡θ,btan⁡θ) equation of the tangent at P is  xasec⁡θ−ybtan⁡θ=1Product of the lengths of the perpendiculars from (esec⁡θ−1)(esec⁡θ+1)sec2⁡θa2+tan2⁡θb2=a2b2e2sec2⁡θ−1a2e2−1sec2⁡θ+tan2⁡θ=b2So statement-1 is trure.For statement-2, by definition of hyperbola distance of P from a focus is e times its distance from the corresponding directrix. So PS=ex−ae and PS′=ex+ae⇒PS−PS′=2aThus statement-2 is also true but does not justify statement-1.