Question

A stright line passes through the points $(\alpha ,\beta )$ and this points bisects the portion of the line intercepted between the axes . Show that the equation of the straight line is$\frac{x}{2\alpha }+\frac{y}{2\beta }=1$

Answer 1

Let the equation of the line be $\frac{x}{a}+\frac{y}{b}=1$

This line cuts the coordinate axes at $A(a,0)$ and $(0,b)$.

It is given that $(\alpha ,\beta )$ bisects the segment $AB$.

$\therefore \alpha =\frac{a+0}{2},\beta =\frac{a+0}{2}\Rightarrow a=2\alpha ,b=2\beta$

Substituting these values in {i}, we get$\frac{x}{2\alpha }+\frac{y}{2\beta }=1$