jmc

We try to rewrite the given equation in one of the standard forms of a conic section. Because both sides are nonnegative, we may square both sides, knowing that this operation is reversible: $$(y+5{)}^2=(x-2{)}^2+y^2.$$Then $$(y+5{)}^2-y^2=(x-2{)}^2,$$or $$10y+25=x^2-4x+4.$$Because there is an $x^2$ term but no $y^2$ term, we recognize that this equation describes a $\boxed{\text{(P)}}$ parabola.