\(x + y =\frac{ d }{4}\)

\(\frac{ A \epsilon_{0}}{ d }= C _{0}\)

\(\Delta V = Ex +\frac{ E }{ k } \times \frac{3 d }{4}+ Ey\)

\(=\frac{3 Ed }{4 k }+ E ( x + y )\)

\(\Delta V = E \left[\frac{3 d }{4 k }+\frac{ d }{4}\right]\)

\(\Delta V =\frac{\sigma}{\epsilon_{0}}\left[\frac{3 d + dk }{4 k }\right]=\frac{ Qd }{ A \epsilon_{0}}\left[\frac{3+ k }{4 k }\right]\)

\(\frac{ Q }{\Delta V }= C =\frac{ A \epsilon_{0}}{ d }\left[\frac{4 k }{3+ k }\right]=\frac{4 k C _{0}}{ k +3}\)