A curve is represented parametrically by the equations `x=t+e^(at) and y=-t+e^(at)` when `t in R and a > 0.` If the curve touches the axis of x at the point A, then the coordinates of the point A are

Options

A. `(1,0)`
B. `(2e,0)`
C. `(e,0)`
D. `(1//e,0)`

Answer

B

Explanation

`x=t+e^(at),y=-t+e^(at)`
`(dx)/(dt)=1+ae^(at), (dy)/(dx)=-1+ae^(at),(dy)/(dx)=(-1+ae^(at))/(1+ae^(at))`
at the point A, `y=0 and (dy)/(dx)=0` for some `t=t_(1)`
`therefore" "ae^(at_(1))=1" (1),"`
`"also "0=-t_(1)+e^(at_(1))," "therefore" "e^(at_(1))=t_(1)`
`"we get, "at_(1)=1 rArr" "t_(1)=(1)/(a),`
now from (i) `ae=1" "rArr" "a=(1)/(e)`
`"hence "x_(A)=t_(1)+e^(at_(1))=e+e=2e`
`rArr" Point A is "(2e,0)`