h[x_,y_]=(((x/5))^2)+(((y/4))^2);

a=RegionPlot[h[x,y]≤1,{x,-7,7},{y,-6,6}];

b=Graphics[{PointSize[Large],Red,Point[{4,5}]}];

g[x_,y_]=(((x-4))^2)+(((y-5))^2);

c=ContourPlot[(Sqrt[g[x,y]]),{x,-7,7},{y,-6,6},ContourShading->False,ContourLabels->True,Contours->20];

Show[a,b,c]

(Sqrt[g[3,3.2]])

(Sqrt[g[-4,-2.3]])

gradg[x_,y_]={D[g[x,y],x],D[g[x,y],y]};

gradh[x_,y_]={D[h[x,y],x],D[h[x,y],y]};

{gradg[x,y]==λ gradh[x,y],h[x,y]==1}

NSolve[%,{x,y,λ}]

(Sqrt[g[2.95799,3.22493]])

(Sqrt[g[-4.03116,-2.36639]])

α[t_]={5 Cos[t],4 Sin[t]};

d=ParametricPlot[α[t],{t,0,2 π}];

Manipulate[Show[b,c,d,Graphics[Line[{{4,5},α[t]}]],Graphics[{PointSize[Large],Red,Point[α[t]]}]],{t,0,2 π}]

e=StreamPlot[gradg[x,y],{x,-7,7},{y,-6,6}];

Show[a,e,c]

Plot[{g[5 Cos[t],4 Sin[t]],0},{t,0 , 2π}]

dist[t_]=Sqrt[g[5 Cos[t],4 Sin[t]]];

Plot[{dist[t],0},{t,0,2 π}]

D[g[5 Cos[t],4 Sin[t]],t]==0

%/.{Cos[t]->u,Sin[t]->v}

NSolve[{%,(u^2)+(v^2)==1},{u,v}]

Sqrt[g[5(0.59159),4(0.80623)]]

Sqrt[g[5(-0.80623),4(-0.59159)]]