1. Determine the equation of the line:
The line passes through the intercepts $(a,0)$ and $(0,b)$. The equation of a line in standard form given intercepts $a$ and $b$ is:
$$\frac{x}{a} + \frac{y}{b} = 1$$
Multiplying through by $ab$ to clear the denominators, we get:
$$bx + ay = ab$$

2. **Calculate the area of rectangle $M$:**
Rectangle $M$ has vertices $(0,0)$, $(x,0)$, $(x,y)$, and $(0,y)$. The area of rectangle $M$ is:
$$\text{Area of } M = x \cdot y = xy$$

3. **Calculate the area of rectangle $N$:**
Rectangle $N$ has vertices $(x,y)$, $(x,b)$, $(a,b)$, and $(a,y)$. The area of rectangle $N$ is:
$$\text{Area of } N = (a - x)(b - y)$$
Expanding this expression, we get:
$$\text{Area of } N = ab - ay - bx + xy$$

4. **Simplify the area of rectangle $N$:**
From the equation of the line $bx + ay = ab$, we can substitute $ab$ with $bx + ay$:
$$\text{Area of } N = ab - ay - bx + xy = (bx + ay) - ay - bx + xy$$
Simplifying further, we see that the terms $bx$ and $ay$ cancel out:
$$\text{Area of } N = xy$$

5. Determine the ratio of the areas:
The ratio of the area of $M$ to the area of $N$ is:
$$\frac{\text{Area of } M}{\text{Area of } N} = \frac{xy}{xy} = 1$$

Conclusion:
$$\boxed{1}$$