Step 1: Understanding the Lassaigne's Test

The sodium fusion extract (Lassaigne's extract) is used to detect elements like nitrogen, sulfur, and halogens in organic compounds. When an organic compound containing both nitrogen and sulfur is fused with sodium, it forms sodium thiocyanate ($NaSCN$).

Step 2: Reaction with Sodium Nitroprusside

When sodium thiocyanate reacts with sodium nitroprusside ($Na_2[Fe(CN)_5NO]$), it produces a blood-red coloration due to the formation of ferric thiocyanate complex ($Fe(SCN)_3$ or related species). This test is specific for the presence of both sulfur and nitrogen in the organic molecule.

Step 3: Analyzing the Amino Acids

We must identify which of the given amino acids contains both sulfur and nitrogen:

• Serine: $C_3H_7NO_3$ (Contains N, but no S)
• Leucine: $C_6H_{13}NO_2$ (Contains N, but no S)
• Threonine: $C_4H_9NO_3$ (Contains N, but no S)
• Methionine: $C_5H_{11}NO_2S$ (Contains both N and S)

Step 4: Conclusion

Since methionine is the only sulfur-containing amino acid among the options, it will yield the blood-red color upon treatment with sodium nitroprusside after sodium fusion.