+194 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.
The point on the graph is of the form (x,21x2−9), so the distance between the origin and this point is x2+(21x2−9)2. Squaring both sides, we get a2=x2+(12x2−9)2 =x2+14x4−9x2+81 =14x4−8x2+81.We can factor this as a2=14(x2−8)2+3244 =14(x2−8)2+81.The minimum value of 41(x2−8)2 is 0, which occurs when x2=8.
Therefore, the minimum value of a^2 is 81, so a^2=81.
