f P is a point on the rectangular hyperbola $x^2-y^2=a^2,C$ is its centre and $S,S^{\mathrm{\prime }}$ are the two foci, then $SP.S^{\mathrm{\prime }}P=$

Let the coordinate of $P\text{ be }(x,y)$

The coordinates of the centre are (0, 0).

The eccentricity of the hyperbola is $\sqrt{1+\frac{a^2}{b^2}}=\sqrt{2}$

So the coordinates of the foci are $S(a\sqrt{2},0)\text{ and }S(-a\sqrt{2},0)$ $\text{ (Fig 19.9) }$

Equation of the corresponding directrices are $x=a/\sqrt{2}$ and $x=-a/\sqrt{2}$

By definition of the hyperbola

$\begin{array}{l}SP=e(\text{ distance of }P\text{ from }x=a/\sqrt{2})\\ =\sqrt{2}|x-a/\sqrt{2}|\end{array}$

Similarly  $S^{\mathrm{\prime }}P=\sqrt{2}|x+a/\sqrt{2}|$

so that $SP.S^{\mathrm{\prime }}.P=2\left|x^2-a^2/2\right|$

$=2x^2-a^2=x^2+y^2=(CP{)}^2$

$\left(\because P\text{ lies on the hyperbola }{x}^2-y^2=a^2\right)$