Let the line segment between axes intersect the x-axis at point $A(x_1, 0)$ and the y-axis at point $B(0, y_1)$.

Since $P(a, b)$ is the mid-point of line segment AB, we apply the mid-point formula:

$$\text{Mid-point of AB} = \left(\frac{x_1 + 0}{2}, \frac{0 + y_1}{2}\right) = \left(\frac{x_1}{2}, \frac{y_1}{2}\right)$$

Given that this mid-point equals $(a, b)$, we get:

$$\frac{x_1}{2} = a \quad \text{and} \quad \frac{y_1}{2} = b$$

Therefore, $x_1 = 2a$ and $y_1 = 2b$.

This means the line makes x-intercept $2a$ and y-intercept $2b$ on the axes.

Using the intercept form of equation of a line:

$$\frac{x}{2a} + \frac{y}{2b} = 1$$

Multiplying both sides by $2$:

$$\frac{x}{a} + \frac{y}{b} = 2$$

∴ Hence proved.