The correct option is D The motion is SHM with amplitude √a2+b2.
From the data given in the question, the displacement
y=asinωt+bcosωt ....(1)
Let a=Acosϕ and b=Asinϕ
Now, a2+b2=A2cos2ϕ+A2sin2ϕ=A2
⇒A=√a2+b2
From (1), we can say that
y=Acosϕ.sinωt+Asinϕ.cosωt=Asin(ωt+ϕ)
Double differentiating with respect to time on both sides, we get
dydt=Aωcos(ωt+ϕ)
d2ydt2=−Aω2sin(ωt+ϕ)=−ω2y
⇒d2ydt2∝(−y)
Hence, it is the equation of an SHM with amplitude A=√a2+b2
Hence, options (a) and (d) are correct.