Question

Locus of the mid-points of the intercepts between the coordinate axes by the lines passing through a fixed point $(a,0)$

Answer 1

Let the equation of line in intercept form be $\frac{x}{l}+\frac{y}{m}=1$
and be $(h,k)$ be the mid point of the intercepts.
Therefore, $(h,k)=(\frac{l}{2},\frac{m}{2})$
given line passes through $(a,0)$
Therefore, $\frac{a}{l}=1\Rightarrow h=\frac{a}{2}$
locus of midpoint is $x=\frac{a}{2}.$