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If |z - 3 + 2i| ≤ 4, then the difference between the greatest value and the least value of |z| is:
- a)4 + √13
- b)2√13
- c)√13
- d)8
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
If |z - 3 + 2i|≤4, then the difference between the greatest value a...
Given equation represents the circle with center(3,−2) and is of radius (R) = 4
∣z∣ represents the distance of point 'z' from origin
Greatest and least distances occur along the normal through the origin
Normal always passes through center of circle
OC = 3 - 2i
|z|min = 0
|z|max = OC + r
OC = 3 - 2i
|z|min = 0
|z|max = OC + r
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If |z - 3 + 2i|≤4, then the difference between the greatest value a...
- Given information:
Given |z - 3 + 2i| ≤ 4.
- Definition of Absolute Value:
The absolute value of a complex number z = a + bi is |z| = sqrt(a^2 + b^2).
- Expressing the Given Inequality:
We can rewrite the given inequality as |z - (3 - 2i)| ≤ 4.
This means that the distance between z and (3 - 2i) is less than or equal to 4.
- Geometric Interpretation:
This inequality represents a circle centered at (3, -2) with a radius of 4 in the complex plane.
- Maximum and Minimum Values:
The maximum value of |z| occurs on the circle when z is farthest away from the center, which is at the point of intersection of the circle with the line passing through the origin and the center of the circle.
The minimum value of |z| occurs when z is closest to the center, which is at the point (3 - 2i).
- Calculating the Values:
The maximum value of |z| can be found by calculating the distance between the point of intersection and the origin. This distance is 4 + sqrt(13).
The minimum value of |z| is the distance between (3 - 2i) and the origin, which is sqrt(13).
- Calculating the Difference:
The difference between the maximum and minimum values of |z| is:
(4 + sqrt(13)) - sqrt(13) = 4 + sqrt(13).
Therefore, the correct answer is option 'A' (4 + sqrt(13)).
Given |z - 3 + 2i| ≤ 4.
- Definition of Absolute Value:
The absolute value of a complex number z = a + bi is |z| = sqrt(a^2 + b^2).
- Expressing the Given Inequality:
We can rewrite the given inequality as |z - (3 - 2i)| ≤ 4.
This means that the distance between z and (3 - 2i) is less than or equal to 4.
- Geometric Interpretation:
This inequality represents a circle centered at (3, -2) with a radius of 4 in the complex plane.
- Maximum and Minimum Values:
The maximum value of |z| occurs on the circle when z is farthest away from the center, which is at the point of intersection of the circle with the line passing through the origin and the center of the circle.
The minimum value of |z| occurs when z is closest to the center, which is at the point (3 - 2i).
- Calculating the Values:
The maximum value of |z| can be found by calculating the distance between the point of intersection and the origin. This distance is 4 + sqrt(13).
The minimum value of |z| is the distance between (3 - 2i) and the origin, which is sqrt(13).
- Calculating the Difference:
The difference between the maximum and minimum values of |z| is:
(4 + sqrt(13)) - sqrt(13) = 4 + sqrt(13).
Therefore, the correct answer is option 'A' (4 + sqrt(13)).
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If |z - 3 + 2i|≤4, then the difference between the greatest value and the least value of |z| is:a)4 +√13b)2√13c)√13d)8Correct answer is option 'A'. Can you explain this answer?
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