Đặt $z_1=x+yi,z_2=c+di\left(x,y,c,d\in ℝ\right).$

Ta có:

$$\begin{gathered} \left|z_1\right|=3\Rightarrow x^2+y^2=9; \\ \left|z_2\right|=4\Rightarrow c^2+d^2=16; \\ \left|z_1-z_2\right|=\sqrt{37}\Rightarrow x^2+y^2+c^2+d^2-2xc-2yd=37\iff xc+yd=-6. \end{gathered}$$

Lại có: $\frac{z_1}{z_2}=\frac{x+yi}{c+di}=\frac{xc+yd}{c^2+d^2}+\frac{yc-xd}{c^2+d^2}i=-\frac{3}{8}+bi.$

Suy ra $a=-\frac{3}{8}.$

Mà $\left|\frac{z_1}{z_2}\right|=\frac{\left|z_1\right|}{\left|z_2\right|}=\frac{3}{4}=\sqrt{a^2+b^2}\iff a^2+b^2=\frac{9}{16}\Rightarrow b^2=\frac{9}{16}-a^2=\frac{27}{64}\Rightarrow b=\pm \frac{3\sqrt{3}}{8}$

Vậy có hai số phức z  thỏa mãn.