If P is a point on the rectangular hyperbola x2−y2=a2, C is its centre and S,S′ are the two foci, the SP⋅S′P=
(CP)2
Let the coordinates of P be (x, y)
The coordinates of the centre C are (0, 0)
The eccentricity of the hyperbola is √1+a2a2=√2
So the coordinates of the foci are S(a√2,0) and S′(−a√2,0).
Equation of the corresponding directrices are x=a√2 and x=−a√2.
By definition of the hyperbola
SP=e⋅(distance of P from x=a√2)=√2∣∣∣x−(a√2)∣∣∣Similarly S′P=√2∣∣∣x+(a√2)∣∣∣⇒ SP⋅S′P=2∣∣x2−a22∣∣=2x2−a2=x2+y2=(CP)2
(∵ P lies on the hyperbola x2−y2=a2)