Find the points on the cone z^2=x^2+y^2 that are closest to

Schwelliney

Schwelliney

Answered question

2021-11-16

Find the points on the cone
z2=x2+y2
that are closest to the point
(2,2,0).

Answer & Explanation

Harr1957

Harr1957

Beginner2021-11-17Added 18 answers

Given,
z2=x2+y2
A point on the cone is defined as P(x, y, z).
So the distance between the point (2,2,0) and P(x,y,z) is
d=(x2)2+(y2)2+(z0)2
d=(x2)2+(y2)2+z2
d=(x2)62+(y2)2+x2+y2
d2=(x2)2+(y2)2+x2+y2 Now,
Since, x2 is an increasing function, minimizing d is same as minimizing f(x,y)=d2 therefore f=0
Now, fx=df dx =2(x2)+2x=0 and fy=df dy =2(y2)+2y=0
2x4x+2x=0 and 2y4+2y=0
x=1 and y=1
So, z=±x2+y2=±12+12=±2
The closest points on the cone are therefore  (2,2,2) and (2,2,2)

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