In[]:=
UseOfOsculatingConic::usage="Use of ResourceFunction, OsculatingConic, for different curves.";
In[]:=
Manipulate[r[t_]:={t,Cosh[t]};δ=ResourceFunction["OsculatingConic"][r[t],{t,t1},{x,y}];​​p=Grid[{{EQUATION},{Simplify[δ]}},FrameAll,ItemSize->12,ItemStyle->12];​​Show[{ParametricPlot[r[t],{t,-4,4},PlotStyle{Thick,Blue}],ContourPlot[δ,{x,-25,25},{y,-25,25},ContourStyle->{Thick,Red},PerformanceGoal"Quality"],Graphics[{Black,PointSize[0.01],Point[r[t1]]}]},AxesTrue,AxesOrigin{0,0},BoxRatiosAutomatic,PlotRange25,ImageSize700],Style["Osculating conic:",Bold,Large],{{t1,-3.9,"Value (t1)"},-4,4,0.01},Delimiter,{{p,2,"Equation of the Osculating Conic"}},ControlPlacementTop,SaveDefinitions->True]
Out[]=
​
Osculating conic:
Value (t1)
Equation of the Osculating Conic
EQUATION
-71.876-47.4869x-16.7079
2
x
+15.7929y+2.67379xy+0.0134696
2
y
0