The number of normal(s) of a rectangular hyperbola which can touch its conjugate hyperbola is equal to 

Normal to hyperbola $xy=c^{2}at\left(ct,\frac{c}{t}\right)isy-\frac{c}{t}=t^2\left(x-ct\right)$

Solving with $xy=-c^2,weget\Rightarrow x\left\{\frac{c}{t}+t^2\left(x-ct\right)\right\}+c^2=0$

$\Rightarrow t^2{x}^2+\left(\frac{c}{t}-c{t}^3\right)x+c^2=0$ since the touches the curve, above equation has equal roots 

$\Rightarrow t^2=\frac{2\pm \sqrt{8}}{2}or\frac{-2\pm \sqrt{8}}{2}\Rightarrow t^2=1+\sqrt{2}or-1+\sqrt{2}$

Thus four such values of ‘t’ are possible.