Check that every subtype has a method defined

Say I have an abstract type with some structs, and I want to make sure some method is defined on every subtype. For example:

abstract type Shape end
struct Square <: Shape

struct Circle <: Shape

area(s::Square) = s.length * s.length
area(c::Circle) = c.radius * c.radius * pi

area_plus(s::Square, y::Float64) = area(s) + y
area_plus(c::Circle, x::Float64, y::Float64) = area(c) + x + y

area_times(s::Square, y::Float) = area(s) * y

Then I’d like to be able to write a function method_exists so that

all_shapes = [Square, Circle]
method_exists(area, all_shapes) # == true
method_exists(area_plus, all_shapes) # == true
method_exists(area_times, all_shapes) # == false

How can I write something that checks for this? hasmethod gets me part of the way there, but it requires the whole signature of the function so I couldn’t find a way to apply it to something like area_plus.

(In real life, we have several dozen “Shapes” and different researchers writing new ones, so I want a quick way to test that each Shape is at least well-formed.)