JEE MAIN - Mathematics (2021 - 26th February Evening Shift - No. 18)
Let z be those complex numbers which satisfy
| z + 5 | $$ \le $$ 4 and z(1 + i) + $$\overline z $$(1 $$-$$ i) $$ \ge $$ $$-$$10, i = $$\sqrt { - 1} $$.
If the maximum value of | z + 1 |2 is $$\alpha$$ + $$\beta$$$$\sqrt 2 $$, then the value of ($$\alpha$$ + $$\beta$$) is ____________.
| z + 5 | $$ \le $$ 4 and z(1 + i) + $$\overline z $$(1 $$-$$ i) $$ \ge $$ $$-$$10, i = $$\sqrt { - 1} $$.
If the maximum value of | z + 1 |2 is $$\alpha$$ + $$\beta$$$$\sqrt 2 $$, then the value of ($$\alpha$$ + $$\beta$$) is ____________.
Answer
48
Explanation
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._26th_February_Evening_Shift_en_18_1.png)
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
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
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