The equation of a circle is given by $$x^2 + y^2 = a^2$$, where $$a$$ is the radius. If the equation is modified to change the origin other than $$(0, 0)$$, then find out the correct dimensions of $$A$$ and $$B$$ in a new equation: $$(x - At)^2 + (y - \frac{t}{B})^2 = a^2$$
The dimensions of $$t$$ is given as $$[T^{-1}]$$

We need to find the dimensions of $$A$$ and $$B$$ in the equation $$(x - At)^2 + \left(y - \frac{t}{B}\right)^2 = a^2$$, where the dimension of $$t$$ is $$[T^{-1}]$$.

Principle of Dimensional Homogeneity: Each term in an equation must have the same dimensions.

Finding dimension of A:

The term $$At$$ must have the same dimension as $$x$$, which is $$[L]$$.

$$[At] = [L]$$

$$[A][T^{-1}] = [L]$$

$$[A] = [LT]$$

Finding dimension of B:

The term $$\frac{t}{B}$$ must have the same dimension as $$y$$, which is $$[L]$$.

$$\left[\frac{t}{B}\right] = [L]$$

$$\frac{[T^{-1}]}{[B]} = [L]$$

$$[B] = \frac{[T^{-1}]}{[L]} = [L^{-1}T^{-1}]$$

Therefore, $$A = [LT]$$ and $$B = [L^{-1}T^{-1}]$$.

The correct answer is Option 2.