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The coordinates of a particle moving in ...

The coordinates of a particle moving in a plane are given by `x=acos pt` and `y=b sin pt ` where `a,b(lta)` and `p` are positive constants of appropriate dimensions. Then,

A

The path of the particle is an ellipse

B

The velocity and acceleration of the particle are normal to each other at `t=pi//2p`

C

The acceleration of the particle is always directed towards a focus.

D

The distance travelled by the particle in time interval `t=0` to `t=pi/(2p)` is a

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`x=a cospt, y=b sin pt, vec(r)=a cos (pt) hat(i)+b sin (pt) hat(j)`
`:' sin^(2)pt+cos^(2)pt=1`
`:. x^(2)/a^(2)+y^(2)/b^(2)=1` (ellipse)
`vec(v)=-ap sin (pt)hat(i)+bp cos (pt)hat(j), v_(t)=pi/(2p)=-aphat(i)`
`vec(a)=-ap^(2)(pt)hat(i)=bp^(2) sin (pt)hat(j), a_(t)=pi/(2p)=-bp^(2)hat(j)`
`vec(a).vec(v)=0`
`vec(a)=-p^(2) [a cos pthat(i)+b sin pthat(j)]=-p^(2)vec(r)`
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