Principle of Homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same. Then,
(x−At)2+(y−Bt)2=a2
For first term,
[At]=[x]⇒A×T1=L
∴[A]=T1L1
For the second term,
[Bt]=[y]
⇒T[B]1=L
⇒[B]=T−1L−1