Number of common tangent with finite slope to the curves $xy=c^{2}and{y}^2=4ax$  is:

Equation of PQ

   $\begin{array}{l}yt-x=a{t}^2-----1\\ \frac{x}{t_1}+y{t}^{\text{'}}=2c-----2\end{array}$                   $\begin{array}{l}\because \left(1\right)and\left(2\right)areidentical\\ \Rightarrow -t^{\text{'}}=\frac{t}{t_1}=\frac{a{t}^2}{2c}\\ \Rightarrow \frac{a{\left(t^{\text{'}}\right)}^4}{2c}=-t^{\text{'}}\\ \Rightarrow {\left(t^{\text{'}}\right)}^3=-\frac{2c}{a}\end{array}$                 
   $\Rightarrow$     Only one real value of t’ exist.