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The least positive vlaue of the paramete...

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.

A

`2`

B

`4`

C

`3`

D

`1`

Text Solution

Verified by Experts

The correct Answer is:
A
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`
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