+35 The smallest distance between the origin and a point on the graph of \(y=\frac{1}{\sqrt{2}}\left(x^2-3\right)\) can be expressed as \(\sqrt{a}/b\), where \(a\) and \(b\) are positive integers such that \(a\) is not divisible by the square of any integer greater than one. Find \(a+b\).
Pls, can I get some pointers on how to even start the question?? Thanks.
By the distance formula, we are trying to minimize\(\sqrt{x^2+y^2}=\sqrt{x^2+(1/2)(x^4-6x^2+9)}\). In general minimization problems like this require calculus, but one optimization method that sometimes works is to try to complete the square. Pulling out a factor of 1/2 from under the radical, we have
\(\begin{align*} \frac{1}{\sqrt{2}}\sqrt{2x^2+x^4-6x^2+9}&=\frac{1}{\sqrt{2}}\sqrt{(x^4-4x^2+4)+5} \\ &= \frac{1}{\sqrt{2}}\sqrt{(x^2-2)^2+5}. \end{align*}\)
This last expression is minimized when the square equals 0, i.e. when x=√2.Then the distance is√5/√2=√10/2 . Hence the desired answer is 12.
