A line passes through $\mathrm{P}(-4,1)$ and meets the co-ordinate axes at points $A$ and $B$. If $P$ divides the segment $A B$ internally in the ratio $1: 2$, then the equation of the line is

The line with intercept form $\frac{x}{a} + \frac{y}{b} = 1$ intersects the axes at $A(a, 0)$ and $B(0, b)$.
Given that $P(-4, 1)$ divides $AB$ internally in the ratio $1:2$, the section formula yields $\frac{2a}{3} = -4$ for the x-coordinate and $\frac{b}{3} = 1$ for the y-coordinate.

Solving gives $a = -6$ and $b = 3$.

Substituting into the intercept form provides $\frac{x}{-6} + \frac{y}{3} = 1$.
Multiplying through by 6 and rearranging yields $x - 2y + 6 = 0$, which corresponds to option A.

Answer: A