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What is the smallest distance between the origin and a point on the graph of y = x^2 - 8.

 Apr 23, 2022
 #1
avatar+61 
-2

I think its when y=0 so \(x^2=8 \\ x=2\sqrt2 \\\)

 

So the answer is \(\boxed{2\sqrt{2}}\)

 Apr 23, 2022
 #2
avatar+124596 
0

Let     ( x, x^2 - 8)   be a point on the graph

 

Using  the square  of the diistance  function we have

 

d ^2  =   (x - 0)^2  +  ( x^2 - 8 - 0)^2

 

d^2  = x^2   +  x^4 - 16x^2 +  64

 

d^2  =  x^4 -  15x^2   +  64             take the derivative of this  function and  set it  = 0

 

4x^3  - 30x = 0

 

x ( 4x^2 - 30)  =0

 

Solving for the second factor

 

4x^2  - 30  = 0

 

x^2  = 30 /4

 

x = sqrt (30)  / 2          and y  = (30/4 - 8)   =  -1/2

 

So....the minimum  distance is    sqrt [ 30/4 + 1/4] =    sqrt (31)  / 2

 

 

cool cool cool

 Apr 23, 2022

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