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

$⇒\left(\frac{x}{2},\frac{y}{2}\right)=(a,b)$

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

$∴x=2a$ 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
$(y-2b)=\frac{\left(0-2b\right)}{\left(2a-0\right)}(x-0)$

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

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

$i.e,bx+ay=2ab$
On dividing both sides by $ab$, we obtain
$\frac{bx}{ab}+\frac{ay}{ab}=\frac{2ab}{ab}$

$⇒\frac{x}{a}+\frac{y}{b}=2$

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