1. **Problem Statement:** Given that point $P(a,b)$ is the midpoint of a line segment between the x-axis and y-axis, show that the equation of the line is $$\frac{x}{a} + \frac{y}{b} = 2.$$ 2. **Understanding the problem:** The line segment intersects the x-axis at some point $(x_1,0)$ and the y-axis at some point $(0,y_1)$. Point $P(a,b)$ is the midpoint of this segment. 3. **Midpoint formula:** The midpoint $P$ of a segment with endpoints $(x_1,0)$ and $(0,y_1)$ is given by: $$P = \left(\frac{x_1 + 0}{2}, \frac{0 + y_1}{2}\right) = \left(\frac{x_1}{2}, \frac{y_1}{2}\right).$$ 4. **Using the given midpoint:** Since $P = (a,b)$, equate coordinates: $$a = \frac{x_1}{2} \implies x_1 = 2a,$$ $$b = \frac{y_1}{2} \implies y_1 = 2b.$$ 5. **Equation of the line using intercept form:** A line crossing the x-axis at $(x_1,0)$ and y-axis at $(0,y_1)$ can be written as: $$\frac{x}{x_1} + \frac{y}{y_1} = 1.$$ 6. **Substitute $x_1$ and $y_1$:** $$\frac{x}{2a} + \frac{y}{2b} = 1.$$ 7. **Multiply both sides by 2:** $$\frac{x}{a} + \frac{y}{b} = 2.$$ **Final answer:** The equation of the line is $$\frac{x}{a} + \frac{y}{b} = 2.$$