Question

A curve is represented parametrically by the equations $x=t+e^{\text {at }}$ and $y=-t+e^{\text {at }}$ 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

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

Answer 1

Answer: (D)

$\frac{ dx }{ dt }=1+ ae ^{ at } ; \frac{ dy }{ dt }=-1+ ae ^{ at } ; \frac{ dy }{ dx }=\frac{-1+ ae ^{ at }}{1+ ae ^{ at }}$
at the point $A, y=0$ and $\frac{d y}{d x}=0$ for some $t=t_1$
$\therefore ac ^{ at _1}=1$....(1)
also $0=- t _1+ e ^{ at _1} ; \therefore e ^{ at _1}= t _1$.....(2),
putting this value in (1)
we get, at $_1=1 \Rightarrow t_1=\frac{1}{a}$;
(1) ae $=1 \Rightarrow a=\frac{1}{e}$
hence $x _{ A }= t _1+ e ^{ at _1}= e + e =2 e \Rightarrow A \equiv(2 e , 0)$