Question

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

Answer 1

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

Since $P(a,b)$ is the midpoint of $AB$
$\{\frac{0+x}{2},\frac{y+0}{2}\}=(a,b)$

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

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

$\therefore x=2a$ and $y=2b$

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

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

$\Rightarrow a(y-2b)=-bx$

$\Rightarrow ay-2ab=-bx$

$\Rightarrow bx+ay=2ab$

On dividing both sides by $ab$, we obtain

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

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

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