Question

P (a, b) is the mid-point of a line segment between axes. Show that equation of the line is ${\displaystyle \frac{x}{a}+\frac{y}{b}=2}$

Answer 1

Let AB be the line segment between the axes and let P (a, b) be its mid-point.

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

Since P (a, b) is the mid-point of AB,

${\displaystyle \frac{0+x}{2},\frac{y+0}{2}=(a,b)}$

=${\displaystyle (\frac{x}{2},\frac{y}{2})=(a,b)}$

= ${\displaystyle \frac{x}{2}=a\text{and}\frac{y}{2}=b}$

∴ x = 2a and y = 2a

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

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

${\displaystyle (y-2b)=\frac{(0-2b)}{(2a-0)}(x-0)}$

${\displaystyle y-2b=\frac{-2b}{2a}\left(x\right)}$

a (y - 2b) = -bx

ay - 2ab = bx

i.e. bx + ay = 2ab

On dividing both sides by ab, we obtain

${\displaystyle \frac{bx}{ab}+\frac{ay}{ab}+\frac{2ab}{ab}}$

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

Thus, the equation of the line is ${\displaystyle \frac{x}{a}+\frac{y}{b}=2}$