Two Points Dividing a Chord in Equal Ratios

As shown in the figure, points $A$ and $B$ lie on the ellipse $\dfrac{x^2}{m} + y^2 = 1$ (where $m > 1$), and $P$ and $Q$ are two distinct points on the line $AB$ satisfying

where $P$ has coordinates $(1, 0)$. As the line $AB$ through $P$ varies, find the equation of the locus of the point $Q$.