Solve d_1^2+d_2^2=4a^2

$$d _ { 1 } ^ { 2 } + d _ { 2 } ^ { 2 } = 4 a ^ { 2 }$$

Solve for d_1 (complex solution)

$$d_{1}=-\sqrt{4a^{2}-d_{2}^{2}}$$

$$d_{1}=\sqrt{4a^{2}-d_{2}^{2}}$$

Solve for a

$$a=\frac{\sqrt{d_{1}^{2}+d_{2}^{2}}}{2}$$

$$a=-\frac{\sqrt{d_{1}^{2}+d_{2}^{2}}}{2}$$

Solve for d_1

$$d_{1}=\sqrt{4a^{2}-d_{2}^{2}}$$

$$d_{1}=-\sqrt{4a^{2}-d_{2}^{2}}\text{, }|d_{2}|\leq 2|a|$$