If P is a point on the hyperbola x2-y2=a2,C is its centre and S,S' are two foci, then SP⋅S'P=

x2-y2=a2,C(0,0)e=2Pasecθ,atanθs2a,0s1−2a,0SP=asecθ−a22+atanθ2SP=a2sec2θ+2−22secθ+tan2θ=a1+2tan2θ+2−22secθSP=a3+2tan2θ−22secθS'P=(asecθ+a2)2+(atanθ)2                     =sec2θ+tan2θ+2+22secθ Step-III:   SPS1P=a29+4tan4θ+12tan2θ-8sec2θ=a24tan4θ+9+4tan2θ−8sec2θ−tan2θ=a22tan2θ+12SPS1P=a21+2tan2θ……(1) Step-IV:   CP=(asecθ)2+(atanθ)2=asec2θ+tan2θ=a1+2tan2θCP2=a21+2tan2θ………(2) From 1)&¯(2)SPS1P=(CP)2