Let AB be a line segment whose midpoint is P (a, b).

Let the coordinates of A and B be (0, y) and (x, 0) respectively.

\( (\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2})\)

\( (\dfrac{0+x}{2}, \dfrac{y+0}{2})\) = (a,b)

\( (\dfrac{x}{2}, \dfrac{y}{2})\) = (a,b)

a = \( \dfrac{x}{2}\) and

b = \( \dfrac{y}{2}\)

x = 2a and y = 2b

Now, the respective coordinates of A and B are (0,2 b) and (2a,0) .

The equation of the line passing through points (0,2b) and (2a,0) is,

\( y-2b=\dfrac{0-2b}{2a-0}(x-0)\)

\( y-2b=\dfrac{-2b}{2a}(x)\)

\( y-2b=\dfrac{-b}{a}(x)\)

a (y – 2b) = -bx

ay – 2ab = -bx

bx + ay = 2ab

Divide both the sides with ab, then

\(\dfrac{bx}{ab}+\dfrac{ay}{ab}=\dfrac{2ab}{ab}\)

\( \dfrac{x}{a}+\dfrac{y}{b}=2\)

Hence proved.

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