$\text{ Let }A\equiv (3,-4),B\equiv (1,2)\text{ . Let }P\equiv (2k-1,2k+1)\text{ be a variable point such that PA + PB is the minimum. Then kis}$

$$\begin{gathered} PA+PB is minimum only when A,B,P are collinear \\ If three points are collinear then the slopes of the lines joining any two points is same \\ PA\hspace{0.17em}slope is \frac{2k+5}{2k-4} and AB slope is \frac{6}{-2} \\ Equate the slopes and then solve for k \\ \frac{2k+5}{2k-4}=-3 \\ 2k+5=-6k+12 \\ 8k=7 \\ k=\frac{7}{8} \end{gathered}$$