{f[t_],g[t_]}={x[t],y[t]}/.NDSolve[{(1/(Sqrt[((x[t])^2)+(((t-y[t]))^2)])){-x[t],t-y[t]}.{x'[t],y'[t]}==1,x'[t]^2+y'[t]^2==1,x[0]==0,y[0]==0},{x[t],y[t]},{t,0,1}][[1]]

((a-p)/(Norm[a-p])).((\[DifferentialD]p)/(\[DifferentialD]t))==2

(Norm[((\[DifferentialD]p)/(\[DifferentialD]t))])==2

p={x[t],y[t]}

a={1,t};p={x[t],y[t]};

((a-p)/(Norm[a-p])).((\[DifferentialD]p)/(\[DifferentialD]t))==2//ComplexExpand

{f[t_],g[t_]}={x[t],y[t]}/.NDSolve[{((x'[t])/(Sqrt[(((1-x[t]))^2)+(((t-y[t]))^2)]))-((x[t] x'[t])/(Sqrt[(((1-x[t]))^2)+(((t-y[t]))^2)]))+((t y'[t])/(Sqrt[(((1-x[t]))^2)+(((t-y[t]))^2)]))-((y[t] y'[t])/(Sqrt[(((1-x[t]))^2)+(((t-y[t]))^2)]))==2,x'[t]^2+y'[t]^2==4,x[0]==0,y[0]==0},{x[t],y[t]},{t,0,1}][[1]]

t1=t/.FindRoot[f[t]==1,{t,0}][[1]]//Chop

Clear[g]

graph1=ParametricPlot[{f[t],g[t]},{t,0,t1},ColorFunction->Function[{x,y},Red]]

FrenetSerretSystem[{f[t],g[t]},t][[1,1]]/.t->0.1

Manipulate[Show[graph1,Graphics[{PointSize[0.05],Red,Tooltip[Point[{f[t],g[t]}],FrenetSerretSystem[{f[s],g[s]},s][[1,1]]/.s->t],Green, Point[{1,t}],Blue, Point[{0.9999999972009388`,0.6666205833775972`}]},Axes->True,PlotRange->{{0,1},{0,2}}]],{t,0,t1},SaveDefinitions->True]

?Tooltip