when the distance from the point (0,−1) is d(x,y) ,
problem is,
d(x,y)=(x−0)2+(y−(−1))2d(x,y)=x2+(y+1)2maximized(x,y)=x2+(y+1)2subject to,x2+16y2=16
Here objective function is d(x,y)=x2+(y+1)2
and constraint is C(x,y)=x2+16y2−16=0
so at the maximum point,
∇d(x,y)=λ∗∇C(x,y)λ is the lagrange multiplier
∇d(x,y)=[2x2(y+1)]
∇C(x,y)=[2x32y]
[2x2(y+1)]=λ[2x32y]
2x=λ∗2xλ=1then,2(y+1)=1∗32yy=151
using C(x,y),x2+16∗(151)2−16=0x=±151614
Therefore furthest points are,
(151614,151) and (−151614,151)
dmax=(151614)2+(151+1)2=45256
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