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The maximum value of |z| when the complex number z satisfies the condition |z + Z/2| = 2 is
- a)√3
- b)√3 +√2
- c)√3 +1
- d)√3 -1
Correct answer is option 'C'. Can you explain this answer?
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The maximum value of |z| when the complex number z satisfies the condi...
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The maximum value of |z| when the complex number z satisfies the condi...
To find the maximum value of |z|, we need to find the complex number z that satisfies |z - Z/2| = 2, where Z is a constant.
Let's assume z = x + yi, where x and y are real numbers.
Substituting this into the equation, we have:
|z - Z/2| = 2
|x + yi - Z/2| = 2
Taking the absolute value of a complex number z = x + yi is equivalent to finding its magnitude, which is given by |z| = sqrt(x^2 + y^2).
So, we need to find the maximum value of sqrt(x^2 + y^2) that satisfies the equation |z - Z/2| = 2.
Expanding the absolute value, we have:
sqrt((x - Z/2)^2 + y^2) = 2
Squaring both sides of the equation, we get:
(x - Z/2)^2 + y^2 = 4
Rearranging the equation, we have:
x^2 - xZ + Z^2/4 + y^2 = 4
Since x^2 + y^2 = |z|^2, we can rewrite the equation as:
|z|^2 - xZ + Z^2/4 = 4
To maximize |z|, we need to minimize xZ. Since Z is a constant, the minimum value of xZ occurs when x = 0.
Substituting x = 0 into the equation, we have:
|z|^2 + Z^2/4 = 4
Simplifying the equation, we get:
|z|^2 = 4 - Z^2/4
To maximize |z|, we need to maximize the right side of the equation. Since Z is a constant, the maximum value of |z| occurs when Z^2/4 is minimized. The minimum value of Z^2/4 occurs when Z = 0.
Therefore, the maximum value of |z| is sqrt(4 - 0) = sqrt(4) = 2.
So, the maximum value of |z| is 2.
Let's assume z = x + yi, where x and y are real numbers.
Substituting this into the equation, we have:
|z - Z/2| = 2
|x + yi - Z/2| = 2
Taking the absolute value of a complex number z = x + yi is equivalent to finding its magnitude, which is given by |z| = sqrt(x^2 + y^2).
So, we need to find the maximum value of sqrt(x^2 + y^2) that satisfies the equation |z - Z/2| = 2.
Expanding the absolute value, we have:
sqrt((x - Z/2)^2 + y^2) = 2
Squaring both sides of the equation, we get:
(x - Z/2)^2 + y^2 = 4
Rearranging the equation, we have:
x^2 - xZ + Z^2/4 + y^2 = 4
Since x^2 + y^2 = |z|^2, we can rewrite the equation as:
|z|^2 - xZ + Z^2/4 = 4
To maximize |z|, we need to minimize xZ. Since Z is a constant, the minimum value of xZ occurs when x = 0.
Substituting x = 0 into the equation, we have:
|z|^2 + Z^2/4 = 4
Simplifying the equation, we get:
|z|^2 = 4 - Z^2/4
To maximize |z|, we need to maximize the right side of the equation. Since Z is a constant, the maximum value of |z| occurs when Z^2/4 is minimized. The minimum value of Z^2/4 occurs when Z = 0.
Therefore, the maximum value of |z| is sqrt(4 - 0) = sqrt(4) = 2.
So, the maximum value of |z| is 2.
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