Electrons in an oscilloscope are deflected by two mutually perpendicular oscillating electric field such that at any time the displacement due to them are given by `x=Acosomegat`, `y=Asin(omegat+pi/6)`. Then the path of the electron is

Options

A. a straight line having the equation `x=y`
B. a circle having the equation
C. an ellipse having the equation
`x^2-sqrt3xy+y^2=A^2/4`
D. an ellipse having the equation
`x^2-xy+y^2=(3A^2)/(4)`

Answer

D

Explanation

`x=Acosomegat`, `y=Asin(omegat+pi//6)`
`y=Asin(omegat+pi/6)=A[sinomegatcos((pi)/(6))+cosomegatsin((pi)/(6))]`
`=A[sqrt3/2sinomegat+1/2cosomegat]`
`=A/2[sqrt3(1-cos^2omegat)^(1/2)+cosomegat]`
`=A/2[sqrt3(1-x^2/A^2)^(1/2)+x/A]`
`(2y)/(A)-x/A=sqrt3(1-x^2/A^2)^(1/2)`
Squaring both sides
`(4y^2)/(A^2)+x^2/A^2-(4xy)/(A^2)=3(1-x^2/A^2)`
`(4y^2)/(A^2)+(4x^2)/(A^2)-(4xy)/(A^2)=3`
`x^2+y^2-xy=(3A^2)/(4)`