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For complex number z, the minimum value of |z| + |z – cos α – i sin α| + |z – 2(cos α + i sin α)| is
- a)1
- b)2
- c)4
- d)Can't say anything
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
For complex number z, the minimum value of |z| + |z – cos &alpha...
|z| + |z - cos α - i sin α| + |z - 2(cos α + i sin α)|
As shown in figure A and B are the points cos α + i sin α and 2(cos α + i sin α) respectively and let z be the affix of P. Then |z| + |z - cos α - i sin α| + |z - 2 (cos a + i sin α.)| is minimum only when P coincides with the point A
∴ OP + AP + BP = 2
As shown in figure A and B are the points cos α + i sin α and 2(cos α + i sin α) respectively and let z be the affix of P. Then |z| + |z - cos α - i sin α| + |z - 2 (cos a + i sin α.)| is minimum only when P coincides with the point A
∴ OP + AP + BP = 2
Most Upvoted Answer
For complex number z, the minimum value of |z| + |z – cos &alpha...
Let z=a+bi where a and b are real numbers.
Then,
|z| = √(a^2 + b^2)
|z+1| = √((a+1)^2 + b^2)
We want to minimize |z| |z+1|, which means we want to minimize the product
|z| |z+1| = √(a^2 + b^2) √((a+1)^2 + b^2)
Squaring both sides, we get:
(|z| |z+1|)^2 = (a^2 + b^2)((a+1)^2 + b^2)
= a^4 + 2a^3 + 3a^2 + 2a + 1 + 2b^2 + 2a^2b^2 + 2ab^2
We can rewrite this expression as:
(|z| |z+1|)^2 = (a^2 + 1)(a^2 + 2a + 2) + 2b^2(1 + a^2)
We notice that the minimum value of the expression is achieved when the first term is minimized, which means when a=0.
Substituting a=0, we get:
(|z| |z+1|)^2 = 1 + 2b^2
The minimum value of this expression is clearly 1, which is achieved when b=0 and a=0.
Therefore, the minimum value of |z| |z+1| is 1, and it is achieved when z=0.
Then,
|z| = √(a^2 + b^2)
|z+1| = √((a+1)^2 + b^2)
We want to minimize |z| |z+1|, which means we want to minimize the product
|z| |z+1| = √(a^2 + b^2) √((a+1)^2 + b^2)
Squaring both sides, we get:
(|z| |z+1|)^2 = (a^2 + b^2)((a+1)^2 + b^2)
= a^4 + 2a^3 + 3a^2 + 2a + 1 + 2b^2 + 2a^2b^2 + 2ab^2
We can rewrite this expression as:
(|z| |z+1|)^2 = (a^2 + 1)(a^2 + 2a + 2) + 2b^2(1 + a^2)
We notice that the minimum value of the expression is achieved when the first term is minimized, which means when a=0.
Substituting a=0, we get:
(|z| |z+1|)^2 = 1 + 2b^2
The minimum value of this expression is clearly 1, which is achieved when b=0 and a=0.
Therefore, the minimum value of |z| |z+1| is 1, and it is achieved when z=0.
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For complex number z, the minimum value of |z| + |z – cos α– i sin α| + |z – 2(cos α+ i sin α)| isa)1b)2c)4d)Can't say anythingCorrect answer is option 'B'. Can you explain this answer?
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