Question

Find the envelope of a straight line which moves so that the sum of the intercepts made by it on two given straight lines is constant.

Answer 1

Let the given lines be axes and let the intercepts made by the line whose envelope is required be $(a-l)$ and $(a+l)$ so that the sum of intercepts is $2a=$ constant ....(given)
Hence the equation of the line will be
$\frac{x}{a-l}+\frac{y}{a+l}=1$
$\Rightarrow l^2+l(x-y)+a(x+y-a)=0$
Hence, the required envelope is obtained by equating the discriminant of $\left(1\right)$ equal to zero, i.e,
${(x-y)}^2-4a(x+y-a)=0$
This represents a parabola touching the axes.