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The coordinates of the point on the curv...

The coordinates of the point on the curve `x^(3)=y(x-a)^(2)` where the ordinate is minimum is

A

`(3a,(27)/(4)a)`

B

`(2a,8a)`

C

`(a,0)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
The ordinate of any point on the curve is given by `y=(x^(3))/((x-1)^(2))`
`(dy)/(dx)=(3x^(2))/((x-1)^(2))-(2x^(3))/((x-1)^(3))=(x^(2)(x-3a))/((x-a)^(3)).`
`(dy)/(dx)=0 rArr x=0 or x = 3a`
`(dy)/(dx)=(3x^(2))/((x-1)^(2))-(2x^(3))/((x-1)^(3))=(x^(2)(x-3a))/((x-a)^(3)).`
`(dy)/(dx)=0 rArr x=0 or x = 3a`

From sign scheme of x = 3a is point of minima and value of y at x = 3a is
`y=((3a)^(3))/((2a)^(2))=(27a^(3))/(4a^(2))=(27)/(4)a`
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