Prove that the Locus of the midpoints of the intercepts between the coordinate axes by the lines passing through (a,0) does not intersect y axis

Question

Let the intercept at y axis be (0,b)

The locus is (h,k)

$$h=\frac a2 , k=\frac b2$$ where b is a parameter

Therefore $$x=\frac a2$$

I have a very crude idea on how it would work. I feel the locus would be parallel to y axis. It might be right, but I can I get a better explanation?

Answer 1

Let line AB intersects X axis at (a,0) and y axis at (0,b). $$ $$ With OA , OB , and AB as sides of triangle , where points O and A are fixed and sides are X axis , Y axis and AB . With point P on AB as midpoint of AB if you join with mid point of OA i.e ($\frac{a}{2}$,0) is parallel to third side i.e Y axis.

Hence locus is X= $\frac{a}{2}$