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Geometry Difficulty 4.9 AIME Find the answer

8. In the complex plane, the point corresponding to the complex number z1z_{1} moves on the line segment connecting 1 and i\mathrm{i}, and the point corresponding to the complex number z2z_{2} moves on the circle centered at the origin with a radius of 1. Then the area of the region where the point corresponding to the complex number z1+z2z_{1}+z_{2} is located is:

Pick one

Solution

8. B.

From the given, z1=t+(1−t)i(0⩽t⩽1)z_{1}=t+(1-t) \mathrm{i}(0 \leqslant t \leqslant 1).
From z2=cos⁡θ+isin⁡θz_{2}=\cos \theta+\mathrm{i} \sin \theta, we get z1+z2=t+cos⁡θ+(1−t+sin⁡θ)iz_{1}+z_{2}=t+\cos \theta+(1-t+\sin \theta) \mathrm{i}.
Let z1+z2=x+yiz_{1}+z_{2}=x+y \mathrm{i}. Then {x=t+cos⁡θ,y=1−t+sin⁡θ. \left\{\begin{array}{l}x=t+\cos \theta, \\ y=1-t+\sin \theta \text {. }\end{array}\right.
Eliminating θ\theta yields (x−t)2+[y−(1−t)]2=1(x-t)^{2}+[y-(1-t)]^{2}=1.
Thus, the point corresponding to z1+z2z_{1}+z_{2} lies on a circle with center (t,1−t)(t, 1-t) (0⩽t⩽1)(0 \leqslant t \leqslant 1) and radius 1.

Therefore, the region where the point corresponding to z1+z2z_{1}+z_{2} lies is the shaded part in Figure 2. Its area is
2⋅π2+22=22+π. \begin{array}{l} 2 \cdot \frac{\pi}{2}+2 \sqrt{2} \\ =2 \sqrt{2}+\pi . \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.