+0  
 
0
105
2
avatar

Find the smallest distance between the origin and a point on the parabola y = x^2 - 2.

 Mar 6, 2022
 #1
avatar+36433 
+1

There may be an easier way to do this....but here is one way

Distance between   origin (0,0)    and    (x, x^2 -2)   using distance formula

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

       = (x^4 -3x^2 +4)1/2

 

 

Now take the derivative to find where the slope of this fxn = 0

     1/2 (4x^3 - 6x) / ( sqrt (x^4-3x^2+4)    = 0

 

          this = 0 when the numerator = 0       2x^3 - 3x   = 0

                                                                    x (2x^2 -3) = 0     means x = 0 or +- sqrt (3/2)

              

Plugging these values of x into the original equation    x^2 - 2     results in coordinate pairs  ( 0, -2)  and  ( sqrt (3/2) , -0.5)  )  (and - sqrt(3/2), - 0.5)

    Now, using these pairs and the origin in the distance formula shows the blue pair(s) result in the shortest distance to the origin

           sqrt ( sqrt(3/2)2  + (-.5)2 ) = d = sqrt ( 7/4) = sqrt (7 ) /2 = 1.32287 units

 Mar 6, 2022
 #2
avatar+36433 
+1

Here is the graph:

 

 

ElectricPavlov  Mar 6, 2022

19 Online Users

avatar