Let z be those complex number which satisfy -z-5- 4 and z1-i-z-1-i-10- i-1- If the maximum value of -z-1-2 is -2- then the value of - is

Let z=x+iy Given, |z+5|≤4 ⇒ (x+5)^2+y^2≤ 16 Also, z(1+i)+z̅(1 i)≥ 10 ⇒ x y ≥ 5....(2) From (1) and (2) Locus of z is the shaded region in the diagram. |z+1| r ...

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Byju's Answer

Standard XI

Mathematics

Geometrical Representation of a Complex Number

Let z be thos...

Question

Let $z$ be those complex number which satisfy $| z + 5 | \leq 4$ and $z (1 + i) + ¯ z (1 - i) \geq - 10, i = \sqrt{- 1}$ . If the maximum value of $| z + 1 |^{2}$ is $α + β \sqrt{2}$ , then the value of $(α + β)$ is

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Solution

Let $z = x + i y$
Given, $| z + 5 | \leq 4$
$\Rightarrow (x + 5)^{2} + y^{2} \leq 16$
Also, $z (1 + i) + ¯ z (1 - i) \geq - 10$
$\Rightarrow x - y \geq - 5.... (2)$
From $(1)$ and $(2)$
Locus of $z$ is the shaded region in the diagram.
$| z + 1 |$ represents distance of ' $z$ ' from $Q (- 1, 0)$
Clearly ' $P$ ' is the required position of ' $z$ ' when $| z + 1 |$ is maximum.
$∴ P \equiv (- 5 - 2 \sqrt{2}, - 2 \sqrt{2})$
$∴ P Q^{2} = 32 + 16 \sqrt{2}$
$\Rightarrow α = 32$
$\Rightarrow β = 16$
Thus, $α + β = 48$

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Let z be those complex number which satisfy |z+5|≤4 and z(1+i)+¯z(1−i)≥−10,i=√−1. If the maximum value of |z+1|2 is α+β√2, then the value of (α+β) is

Let $z$ be those complex number which satisfy $| z + 5 | \leq 4$ and $z (1 + i) + ¯ z (1 - i) \geq - 10, i = \sqrt{- 1}$ . If the maximum value of $| z + 1 |^{2}$ is $α + β \sqrt{2}$ , then the value of $(α + β)$ is