Modulus of a complex number depending on a parameter

Question

How to solve |2 + t + t*i|$^2$? I know the formula for z = x + iy, then |z| = $\sqrt{x^2 + y^2}$

Answer 1

For complex $t$ we have $$z:=2+(1+i)t\implies |z|^2=zz^\ast=(2+(1+i)t)(2+(1-i)t^\ast)\\=4+2tt^\ast+2(t+t^\ast + i(t-t^\ast))=4+2|t|^2+4(\Re t-\Im t).$$