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What is the smallest distance between the origin and a point on the graph of y = x^2 - 3?

 Apr 24, 2021
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Let  the point  be (  x , x^2  -3)

 

Using the  square of the  the  distance formula, we have

 

D^2  =  (x  -0)^2   +  (  x^2 - 3 - 0)^2           simplify

 

D'2  =  x^2   +  x^4  - 6x^2  +  9

 

D^2  =  x^4  - 5x^2  +  9

 

Take the derivative  of   D^2   and set to 0

 

4x^3   -  10x   =  0      factor

 

x ( 4x^2  - 10)  =  0

 

The second factor  set  to  0   gives  us what we need

 

4x^2 -  10  =  0 

 

2x^2  - 5 = 0

 

2x^2  = 5

 

x^2   = 5/2      due to symmetry.....there will  be  two points of equal distance...so....we  can take the + root

 

x = sqrt (5/2)

 

And  y  =  X^2   -  3   =  5/2  -3  = -1/2

 

So  the  point  is    (sqrt (5/2) , -1/ 2 )

 

And  the disance is    sqrt  ( 5/2  + (-1/2)^2 )    = sqrt  ( 11/4)

 

See the graph here  :  https://www.desmos.com/calculator/47lpdp6t98

 

 

cool cool cool      

 Apr 24, 2021

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