Start from the bug, not the stone.
The examiner gives you one decisive word: Proteus. Proteus mirabilis is the prototype UREASE-positive uropathogen, and urease is the enzyme that links infection to a specific stone chemistry. So the question is really "which stone does an alkaline, urease-driven urine make?"
Trace the chemistry.
Urease hydrolyses urea to ammonia and carbon dioxide:
\[\text{Urea} \xrightarrow{\text{urease}} 2\,\text{NH}_3 + \text{CO}_2\]
The liberated ammonia drives urinary pH up. Alkaline urine is supersaturated for magnesium ammonium phosphate (struvite) and for carbonate-apatite (calcium phosphate), so these phosphate salts crystallise out and can build into a large, often staghorn or bladder, infection stone that is radio-opaque on plain film - matching the X-ray finding.
Map to the options.
Only one listed salt is a phosphate that precipitates in this alkaline, infection setting: calcium phosphate. The other three each belong to a different, non-infective mechanism:
$\bullet$ Cystine - inherited cystinuria, acidic urine, hexagonal crystals.
$\bullet$ Calcium oxalate - the commonest stone, but driven by metabolic/dietary factors in acidic urine, not by urease.
$\bullet$ Xanthine - rare enzyme defect/drug-related, radiolucent.
These struvite/phosphate stones precipitate fast in the supersaturated alkaline urine, are radio-opaque, and characteristically enlarge into bulky staghorn or bladder calculi exactly as described in the stem. Therefore the Proteus (urease) stone here is calcium phosphate (A).