Step-by-step Solution:

The median will pass through \(A\) and the midpoint of the opposite side, i.e., \((p, q)\). The general equation of the line that passes through the intersection point of the lines \(AC\) and \(AB\) is: \[ (px + qy - 1) + \lambda (qx + py - 1) = 0 \] As this line passes through the point \((p, q)\), hence: \[ (p^2 + q^2 - 1) + \lambda (2pq - 1) = 0 \Rightarrow \lambda = - \frac{p^2 + q^2 - 1}{(2pq - 1)} \] Putting this value of \(\lambda\), we get the desired equation: \[ (px + qy - 1) - \left( \frac{p^2 + q^2 - 1}{2pq - 1} \right)(qx + py - 1) = 0 \] Or \[ (2pq - 1)(px + qy - 1) - (p^2 + q^2 - 1)(qx + py - 1) = 0 \]