The composition of two simple harmonic motions of equal periods at right angle to  each other and with a phase difference of π results in the displacement of the   particle along

Let us consider two simple harmonic motion y=bsinωt ……………………..(1)x=asinωt+ϕ ………………………(2)x=asinωtcosϕ+acosωtsinϕ  From (1) yb=sinωt  and 1−y2b2=cos   ωt Substituting these is equation (2) we get xa=ybcosϕ+1−y2b2sinϕ Taking square on both sides we get  xa−ybcosϕ2=1−y2b2sinϕ2Thus xa2+yb2cos2ϕ−2xyabcosϕ=1−y2b2sin2ϕ Simplifying  this we getxa2+yb2−2xyabcosϕ=sin2ϕIn this case   xa2+yb2+2xyabcosϕ=0 xa+yb2=0 Simplifying we get  y=−xabMaking the equation a straight line