Question

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

Answer 1

The equation of the line with intercepts a and b is $\frac{x}{a}+\frac{y}{b}=1$

This line intersects the axes at A (a, 0) and B (0, b).

Here, (α, β) is the midpoint of AB.

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

Hence, the equation of the line is $\frac{x}{2\mathrm{\alpha }}+\frac{y}{2\mathrm{\beta }}=1$