QJEEM21Ma41be: JEE Main 2021 Mathematics - Class 11 JEEM21Ma41be Solution: Let z be those complex numbers which satisfy

Let, z = x + iy

Given, z(1 + i) + $\overline z$ (1 $-$ i) $\ge$ $-$ 10

$\Rightarrow$ z + $\overline z$ + i (z $-$ $\overline z$ ) $\ge$ $-$ 10

$\Rightarrow$ 2x + i (2iy) $\ge$ $-$ 10

$\Rightarrow$ x + i² y $\ge$ $-$ 5

$\Rightarrow$ x $-$ y $\ge$ $-$ 5 ...... (1)

Also given, | z + 5 | $\le$ 4

$\Rightarrow$ | z $-$ ( $-$ 5 + 0i) | $\le$ 4 ...... (2)

It represents a circle whose center at ( $-$ 5, 0) and radius 4. z is inside of the circle.

From (1) and (2) z is the shaded region of the diagram.

Now, | z + 1 | = | z $-$ ( $-$ 1 + 0 i) | = distance of z from ( $-$ 1, 0).

Clearly 'p' is the required position of 'z' when | z + 1 | is maximum.

$\therefore$ P $\equiv$ ( $-$ 5 $-$ 4 cos45 $^\circ$ , 0 $-$ 4sin45 $^\circ$ ) = ( $-$ 5 $-$ 2 $\sqrt 2$ , $-$ 2 $\sqrt 2$ )

$\therefore$ (PQ)²|_max = 32 + 16 $\sqrt 2$

$\Rightarrow$ $\alpha$ = 32

$\Rightarrow$ $\beta$ = 16

Thus, $\alpha$ + $\beta$ = 48