given a straight line $y-6=m(x-4)$ passing through $(4,6)$ and meets $xy=4$ at two points, $P$ and $Q$. The midpoint of $P$ and $Q$ is $(\cfrac{2m-3}{m}, 3-2m)$. Find the locus of the midpoint of all the chords that pass through $(4,6)$.
This is what i have tried,
let $N$ be the Midpoint $(x_n,y_n)=(\cfrac{2m-3}{m}, 3-2m)$, $P=(x_p,y_p)$ and $Q=(x_q,y_q)$
By substitution and solving the $x$ and $y$, I get $P$ and $Q$ be $(\cfrac{4}{3-2m\pm \sqrt{(2m-3)^2 +4m)}},3-2m\pm \sqrt{(2m-3)^2 +4m} )$
then for the locus, $$PN=NQ$$ $$\sqrt{(x_p-x_n)^2+(y_p-y_n)^2}=\sqrt{(x_q-x_n)^2+(y_q-y_n)^2}$$ $$(x_p-x_n)^2+(y_p-y_n)^2=(x_q-x_n)^2+(y_q-y_n)^2$$ $$(x_p-x_n)^2-(x_q-x_n)^2=(y_q-y_n)^2-(y_p-y_n)^2$$ $$[(x_p-x_n)+(x_q-x_n)][(x_p-x_n)-(x_q-x_n)]=[(y_q-y_n)+(y_p-y_n)][(y_q-y_n)-(y_p-y_n)]$$ $$(x_p+x_q-2x_n)(x_p-x_q)=(y_q+y_p-2y_n)(y_q-y_p)$$ but i noticed that $x_p+x_q-2x_n=0$, so i am stucked at this point, not sure where i am doing mistake. I hope someone can guide me.