"My mother cooks food on a burner."
Translation:Min mor laver mad på en kogeplade.
I think it depends on where you're from. To me (in AZ in theUS), burner sounds fine for either gas or electric, and "cooktop" more strongly indicates the whole top of the stove than an individual heating element.
"Hob" is pretty thoroughly a Britishism and "hotplate" would refer to an individual, standalone cooking surface not attached to a stove.