Turing reduction problem, intersection is equal to an empty set
Assume $L_c$ is decidable. We will reduce $A_{TM} =\{x\mid M_x \text{ halts on } x\}$ (the Halting problem language) to $L_c$. On input $x $ create a TM $M'$ which halts on only $x$. Then $L(M') = \{x\}$ and $M_x$ halts on $x$ iff $L(M_x) \cap L(M') = \{x\} \neq \emptyset$ iff $\left(\langle M_x \rangle, \langle M' \rangle \right) \notin L_c$. Thus, to check if $M_x$ halts on $x$ we check if $\left(\langle M_x \rangle, \langle M' \rangle \right) \notin L_c$.