The resultant of two rectangular simple harmonic motionsx=asinωt+ϕ and y=bsinωt of the same frequency   and unequal amplitudes but differing in phase by π2  is

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   ωtSubstituting these is equation (2) we get xa=ybcosϕ+1−y2b2sinϕ Taking square on both sides we get xa−ybcosϕ2=1−y2b2sinϕ2 Thus xa2+yb2cos2ϕ−2xyabcosϕ=1−y2b2sin2ϕSimplifying  this we getxa2+yb2−2xyabcosϕ=sin2ϕIn this case   ϕ=π2Making the equation an ellipse