Consult the convexity of the expression

Dear professor:
I have recently been dealing with state constraint with the expression as follows:
a’x + sqrt(a’Pa) <= b (1)
where the vector x and positive definite matrix P are decision variables, and a, b are constant vectors. In order to convex (1) without direct linearization, I introduce an augmented variable r, and transform (1) into following expression:
a’x + r<= b (2)
a’Pa - r^2 <=0 (3)
r > 0 (4)
I want to consult whether the expression (3) is convex and whether this transformation is appropriate.
Thank you very much.

The transformation is appropriate but the expression (3) is not convex. See a special case p-r^2<=0. Draw this set,you’ll find it is not a convex set: