If a curve is represented parametrically by the equation `x = 4t^(3)+3` and `y=4+3t^(4)` and `(((d^(2) x)/(dy^(2))))/(((dx)/(dy))^(n))` is constant then value of `n/2` is

Options

A. 3
B. 4
C. 5
D. 6

Answer

C

Explanation

We have, `x=4t^(3)+3" and "y=4+3t^(4)`
`(dx)/(dt)=12t^(2)" and "(dy)/(dt)=12t^(3)`
`:." "(dy)/(dx)=(dy//dt)/(dx//dt)=(12t^(3))/(12t^(2))=t`
`implies" "(dx)/(dy)=(1)/(t)=t^(-1)`
`implies" "(d^(2)x)/(dy^(2))=-t^(-2)(dt)/(dy)=-t^(-2)xx(1)/(12t^(3))=(1)/(12)t^(-5)`
`:." "(d^(2)x)/(dy^(2))/((dx)/(dy))^(n)=-(1)/(12t^(5))xxt^(n)=-(1)/(12)t^(n-5)`
It is given that `(d^(2)x)/(dy^(2))/((dx)/(dy))^(n)` is constant. Therefore, n=5.