The coordinate of a particle moving in a plane are given by ` x(t) = a cos (pt) and y(t) = b sin (pt)` where `a,b (lt a)` and `P` are positive constants of appropriate dimensions . Then

Options

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

Answer

A, B, C

Explanation

`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)`