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Mathematics Question on Relations and functions

Let S=z∈C:∣z−2∣≤1,and(1+i)+z‾(1−i)≤2.S={z∈C:|z−2|≤1,and(1+i)+\overline{z}(1−i)≤2}. Let ∣z−4i∣|z−4i| attains minimum and maximum values, respectively, at z1∈s and z2∈s. If 5(∣z1∣2+∣z2∣2=α+β√55(|z_1|^2+|z_2|^2=α+β√5 where α and β are integers, then the value of α+β is equal to _________.

Answer

The correct answer is: 26

Let S={z∈C:|z−2|≤1,and1+i+z¯1−i≤2}. Let |z−4i| attains minimum and maximum values, respectively, at z1∈s and z2∈s.

S represents the shaded region shown in the diagram.

Clearly, z 1 will be the point of intersection of PA and given circle.

PA : 2 x + y = 4 and given circle has equation

(x – 2)2 + y 2 = 1.

On solving, we get

z1=()+25i=∣z1∣2=5−45z_1=()+\frac{2}{5}i=|z_1|^2=5-\frac{4}{\sqrt5}

z 2 will be either B or C.

So

5(∣z1∣2+∣z2∣2=30−4√55(|z_1|^2+|z_2|^2=30-4√5

Clearly α = 30 and β = –4 ⇒α + β = 26