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Question

The velocity of a particle moving in the xy plane is given by dxdt=4πsin(3πt) and dydt=3πcos(3πt). If the particle starts from origin at t=0 then the path of the particle is a/an

A
straight line
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B
hyperbola
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C
ellipse
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D
parabola
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Solution

The correct option is C ellipse
For motion along x axis,
dxdt=4πsin(3πt)
dx=4πsin(3πt)dt
On integrating it with limit, x=0 to x and t=0 to t, we get,
x=43[1cos(3πt)]
cos(3πt)=13x4 .............(1)
For motion along y axis,
dydt=3πcos(3πt)
dy=3πcos(3πt)dt
On integrating it with limit, y=0 to y and t=0 to t, we get,
y=sin(3πt)
sin(3πt)=y .............(2)
We know that, sin2θ+cos2θ=1.
From (1) and (2),
y2+(13x4)2=1
This equation is similar to the equation of an ellipse, x2A2+y2B2=1

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