Home
Class 12
MATHS
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`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • TEST PAPERS

    RESONANCE ENGLISH|Exercise Math|105 Videos

  • TEST PAPER

    RESONANCE ENGLISH|Exercise MATHEMATICS|48 Videos

  • TEST SERIES

    RESONANCE ENGLISH|Exercise MATHEMATICS|132 Videos

Similar Questions

Explore conceptually related problems

The least positive integral value of 'k' for which there exists at least one line that the tangent to the graph of the curve y=x^(3)-kx at one point and normal to the graph at another point is

If sum _( k =1) ^(oo) (k^(2))/(3 ^(k))=p/q, where p and q are relatively prime positive integers. Find the value of (p-q),

If sum _( k =1) ^(oo) (k^(2))/(3 ^(k))=p/q, where p and q are relatively prime positive integers. Find the value of (p+q),

The set of points (x,y) in the plane satisfying x ^(2//5)+ |y| =1 form a curve enclosing a region of area p/q square units, when p and q are relatively prime positive intergers. Find p-q.

Find the point on the curve 9y^2=x^3, where the normal to the curve makes equal intercepts on the axes.

India and Australia play a series of 7 one-day matches. Each team has equal probability of winning a match. No match ends in a draw. If the probability that India wins atleast three consecutive matches can be expressed as (p)/(q) where p and q are relatively prime positive integers. Find the unit digit of p.

Three different numbers are selected at random from the set A={1,2,3,……..,10} . Then the probability that the product of two numbers equal to the third number is (p)/(q) , where p and q are relatively prime positive integers then the value of (p+q) is :

For the parabola y=-x^(2) , let a lt 0 and b gt 0,P(a, -a^(2)) and Q(b, -b^(2)) . Let M be the mid-point of PQ and R be the point of intersection of the vertical line through M, with the parabola. If the ratio of the area of the region bounded by the parabola and the line segment PQ to the area of the triangle PQR be (lambda)/(mu) , where lambda and mu are relatively prime positive integers, then find the value of (lambda+mu) :

Find the points on the curve 9y^2=x^3 where normal to the curve makes equal intercepts with the axes.

Find the points on the curve 9y^2=x^3 where normal to the curve makes equal intercepts with the axes.

Home
Profile