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← JEE Main 2022 · 29 Jun · Shift 1
MathematicsInteger+4 / -1Question 21 of 90

JEE Main 2022 · 29 Jun · Shift 1, Mathematics question 21

Let S={z∈C:∣z−2∣≤1, z(1+i)+zˉ(1−i)≤2}S=\left\{z\in\mathbf C:|z-2|\le1,\ z(1+i)+\bar z(1-i)\le2\right\}. Let ∣z−4i∣|z-4i| attains minimum and maximum values, respectively, at z1∈Sz_1\in S and z2∈Sz_2\in S. If 5(∣z1∣2+∣z2∣2)=α+β55\left(|z_1|^2+|z_2|^2\right)=\alpha+\beta\sqrt5, where α\alpha and β\beta are integers, then the value of α+β\alpha+\beta is equal to ______.

Answer and worked solution

Official answer: 26

S is the part of the disc ∣z−2∣≤1|z-2|\le1 with x−y≤1x-y\le1.

Nearest to 4i4i. The disc point toward 4i4i: z1=2−15+2i5z_1=2-\frac1{\sqrt5}+\frac{2i}{\sqrt5} (it satisfies x−y≤1x-y\le1), ∣z1∣2=5−45|z_1|^2=5-\frac4{\sqrt5}.

Farthest. z2=1z_2=1 (end of the cut, ∣z2∣2=1|z_2|^2=1).

Value. 5(6−45)=30−455(6-\frac4{\sqrt5})=30-4\sqrt5: α+β=26\alpha+\beta=26.

Answer: 26

Topics:Geometry in the complex plane
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