What are the dimensions of a and b in the relation: F =a + bx, where F is force and x is distance?

The correct option is MT^-2 (which corresponds to the dimensional formula of the quantity $b$).

According to the principle of homogeneity of dimensions, the dimensions of all terms on both sides of a physical equation must be identical. This means that we can only add or subtract physical quantities if they have the same dimensions, and their result must also share that same dimension.

Given the relation:
$F=a+bx$
where:
- $F$ represents force, which has the dimensional formula:
$\left[F\right]=\left[ML{T}^{-2}\right]$
- $x$ represents distance, which has the dimensional formula:
$\left[x\right]=\left[L\right]$

By applying the principle of homogeneity, the dimensions of the term $a$ must be equal to the dimensions of force $F$:
$\left[a\right]=\left[F\right]=\left[ML{T}^{-2}\right]$

Similarly, the dimensions of the term $bx$ must also equal the dimensions of force $F$:
$\left[bx\right]=\left[F\right]$
This can be written as:
$\left[b\right]\left[x\right]=\left[F\right]$
Substituting the known dimensions of $x$ and $F$:
$\left[b\right]\left[L\right]=\left[ML{T}^{-2}\right]$
Solving for the dimensions of $b$:
$\left[b\right]=\frac{\left[ML{T}^{-2}\right]}{\left[L\right]}$
$\left[b\right]=\left[M{T}^{-2}\right]$

Therefore, the dimensional formula of $a$ is $\left[ML{T}^{-2}\right]$ and that of $b$ is $\left[M{T}^{-2}\right]$, which corresponds to the correct option MT^-2.