The least positive vlaue of the parameter 'a' for which there exist atleast one line that is tangent to the graph of the curve `y= x ^(3)-ax,` at one point and normal to the graph at another point is `p/q,` where p and q ar relatively prime positive integers. Find product pq.

Options

A. `2`
B. `4`
C. `3`
D. `1`

Answer

A

Explanation

Tangent at `(t,t^(3)-kt)` is `y=(3t^(2)-k)x-2t^(3)`
This line cuts the curve again at `x=-2t` where slope of tangent is `12t^(2)-k`
`implies(12t^(2)-k)(3t^(2)-k)=-1`
for positive root `Dge0implies9(3k-4)(3k+4)ge0`
`kepsilon(4/2,oo)implies` Ans `2`