Locus of the mid-points of the intercepts between the coordinate axes by the lines passing through  (a, 0) does not intersect

Equation of any line through (a, 0)  be xa+yb=1 where b is a parameter. This line meets y-axis t (0, b) and if (h, k) denotes the mid-point of the intercept of the line between the coordinate  axes, then h=a / 2, k= b / 2 and then the locus of (h,k) is x=a/2. This clearly does  not intersect y-axis.