The smallest distance between the origin and a point on the graph of y = $\frac{1}{2}x^2$ - 9 can be expressed as a. Find a^2.
+128407 \( y = $\frac{1}{2}x^2$ - 9 \)
Call the point we are looking for (c , c^2/2 - 9 )
Using the square of the distance formula we have
D^2 = ( c - 0)^2 + ( c/^2 / 2 - 9 - 0 )^2
D^2 = c^2 + c^4/4 - 9c^2 + 81
D^2 = c^4 /4 - 8c^2 + 81 take the derivative
D^2 ' = c^3 -16c set to 0
c^3 - 16c = 0
c ( c^2 - 16) = 0
The second factor set = 0 is what we want
c^2 = 16
Two vaues of c will minimize the distance c = 4 or c =-4......choose c = 4
And y = 4^2 / 2 - 9 = -1
The point that minimizes the distance is ( 4 , -1)
a^2 = 4^2 + (-1)^2 = 17
