The set of points on the axis of the parabola $2({(x-1)}^2+{(y-1)}^2)={(x+y)}^2,$  from which 3 distinct normals can be drawn to the parabola, is the set of points (h, k) lying on the axis of the parabola such that.

The directrix of the parabola is $x+y=0$  and focus is (1, 1).
The length of perpendicular from (1, 1) to the directrix is  $\sqrt{2}\Rightarrow \text{\hspace{0.17em}focal\hspace{0.17em}length\hspace{0.17em}is\hspace{0.17em}}\frac{1}{\sqrt{2}}$
The point on the axis which is at a distance equal to focal length from the focus towards the right is (3/2, 3/2). All the points to the right of this point, lying on the axis will satisfy the required condition.