3x+4y+5=0, y=-(3x+5)/4.
|z|=|x+iy|=√(x²+y²)=√(x²+(9x²+30x+25)/4).
|z|=√(13x²+30x+25)/2.
Let v²=(13x²+30x+25)/4,
Differentiate:
2vdv/dx=(26x+30)/4=(13x+15)/2.
When dv/dx=0 there is an extremum, so x=-15/13.
y=-(3x+5)/4=-(-45/13+5)/4=-5/13.
|x+iy|=√(225/169+25/169)=√250/13=1.2163 approx.