**Core Concept**
Penicillin-resistant gonorrhoea is a type of bacterial infection caused by Neisseria gonorrhoeae that exhibits resistance to penicillin antibiotics. This resistance is often mediated through genetic mutations or the acquisition of resistance genes.

**Why the Correct Answer is Right**
The treatment of choice for penicillin-resistant gonorrhoea is ceftriaxone, a third-generation cephalosporin antibiotic. Ceftriaxone works by inhibiting cell wall synthesis in bacteria, ultimately leading to bacterial cell death. This mechanism is specific to beta-lactam antibiotics like ceftriaxone, which target the penicillin-binding proteins (PBPs) in the bacterial cell wall.

**Why Each Wrong Option is Incorrect**
**Option A:** This option is incorrect because amoxicillin is a penicillin antibiotic that would be ineffective against penicillin-resistant gonorrhoea.
**Option B:** This option is incorrect because doxycycline is an antibiotic that targets different bacterial pathways and is not the primary treatment for gonorrhoea.
**Option C:** This option is incorrect because azithromycin is often used in combination with ceftriaxone for the treatment of gonorrhoea, but it is not the sole treatment for penicillin-resistant cases.

**Clinical Pearl / High-Yield Fact**
It's essential to note that the Centers for Disease Control and Prevention (CDC) recommend dual antibiotic therapy for the treatment of gonorrhoea, including ceftriaxone and azithromycin, to address the increasing prevalence of antibiotic-resistant strains.

**Correct Answer: A. Ceftriaxone**