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Question

A curve is represented parametrically by x=t+eat and y=t+eat, tR and a>0. Then, the curve touches the x axis at

A
(1,0)
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B
(1e,0)
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C
(e,0)
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D
(2e,0)
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Solution

The correct option is D (2e,0)
Given curve is x=t+eat; y=t+eat
Let the curve touches x axis at (x1,0), then the slope of the tangent at this point is 0,
Now,
dxdt=1+aeatdydt=1+aeatdydx=1+aeat1+aeat
At (x1,0),
1+aeat11+aeat1=0aeat1=1(1)
Also,
y=0t1+eat1=0eat1=t1
Using equation (1), we get
t1=1a
Again using equation (1), we get
ae=1a=1et1=ex1=t1+eat1x1=e+e=2e
Hence, the point of contact is (2e,0)

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