Question

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

Answer 1

$\text{The equation of the line with intercepts}\text{a and b is}\frac{x}{a}+\frac{y}{b}=1\text{This line intersects the axes at A (a, 0) and B (0, b)}\text{Here,}(\alpha ,\beta )\text{is the midpoint of AB}\therefore \alpha =\frac{a+0}{2},\beta =\frac{0+b}{2}\Rightarrow a=2\alpha ,b=2\beta \text{Hence, the equation of the line is}\frac{x}{2\alpha }+\frac{y}{2\beta }=1$