Step 1: Read the key sign.
Xanthomas located on the Achilles TENDON (and other extensor tendons) are tendon xanthomas. The cell type and location of a xanthoma tell you which lipid is in excess, so use the site to back-track to the disorder.
Step 2: Link tendon xanthoma to its lipid.
Tendon xanthomas are cholesterol-laden deposits and are the hallmark of chronically very high LDL cholesterol. Among the options, the disorder defined by grossly elevated LDL is Familial hypercholesterolaemia (an LDL-receptor / ApoB defect, type IIa pattern).
Step 3: Cross out by matching each disease to its own xanthoma/lipid.
$\bullet$ Tangier disease: deficient HDL with orange tonsils and low cholesterol $-$ no tendon xanthomas.
$\bullet$ Familial hyperchylomicronaemia (type I, LPL deficiency): very high triglycerides giving ERUPTIVE xanthomas, not tendon xanthomas.
$\bullet$ Type III hyperlipoproteinaemia (dysbetalipoproteinaemia): raised IDL/remnants giving PALMAR (palmar crease) and tuberoeruptive xanthomas, not Achilles tendon ones.
Step 4: Confirm the match.
Only familial hypercholesterolaemia couples high LDL with the cholesterol-rich tendon xanthomas seen on the Achilles tendon, and it also explains early corneal arcus and premature coronary disease.
Step 5: Conclusion.
Achilles tendon xanthomas point to LDL excess, i.e. Familial hypercholesterolaemia.
Final answer: Option 1 - Familial hypercholesterolemia.