$P$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1S$ $S\text{ and }{S}^{\mathrm{\prime }}$ are its foci. 

Statement-1: Product of the lengths of the perpendiculars from S and $S^{\mathrm{\prime }}$ on the tangent at P is equal to 

Statement-2: $\left|PS-P{S}^{\mathrm{\prime }}\right|=2a$.

 

detailed solution

Correct option is B

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.