Step 1: Place Tay-Sachs in the family of sphingolipidoses. It belongs to the GM2 gangliosidoses, where the lipid that fails to be broken down is the GM2 ganglioside.
Step 2: The catabolic enzyme for GM2 is hexosaminidase A (the alpha subunit defect). When it is absent the ganglioside builds up inside neuronal lysosomes, the child shows worsening neurologic decline, a heightened startle to sound, blindness and a cherry-red macular spot, with the liver and spleen staying normal in size.
Step 3: Eliminate the rest by their own enzymes. Hexosaminidase B alone is not the classic Tay-Sachs defect; loss of both A and B is Sandhoff disease. $Sphingomyelinase$ loss gives Niemann-Pick, and $\alpha$-galactosidase loss gives Fabry disease.
\[\boxed{\text{Hexosaminidase A}}\]