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For all complex numbers z1 ,z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5 , the minimum value of |z1 -z2| is
- a)0
- b)7
- c)2
- d)17
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
For all complex numbers z1 ,z2 satisfying |z1| = 12 and |z2 - 3 - 4i| ...
Minimum value of |z1 - z2| = 12 - 10 = 2
Most Upvoted Answer
For all complex numbers z1 ,z2 satisfying |z1| = 12 and |z2 - 3 - 4i| ...
Understanding the Problem
To find the minimum value of |z1 - z2| given the conditions on z1 and z2, we need to analyze the geometric representation of these complex numbers.
Conditions Given
- |z1| = 12: This means z1 lies on a circle centered at the origin (0, 0) with a radius of 12.
- |z2 - 3 - 4i| = 5: This indicates that z2 lies on a circle centered at (3, 4) with a radius of 5.
Geometric Interpretation
- The first circle (for z1) is centered at (0, 0) with a radius of 12.
- The second circle (for z2) is centered at (3, 4) with a radius of 5.
Finding the Minimum Distance
1. Distance Between Centers:
- Calculate the distance between the centers of the two circles:
- Distance = sqrt((3 - 0)² + (4 - 0)²) = sqrt(9 + 16) = sqrt(25) = 5.
2. Minimum Distance Calculation:
- The minimum distance between any point on circle z1 and any point on circle z2 occurs when we move from the edge of z1 directly towards the edge of z2.
- This is calculated as: Distance between centers - radius of z2 - radius of z1.
- Minimum Distance = 5 - 5 - 12 = -12 (not valid).
- However, we need to consider the edges of the circles. The effective distance we need is:
- Minimum Distance = Distance between centers + radius of z1 - radius of z2 = 5 + 12 - 5 = 12.
3. Minimum Value of |z1 - z2|:
- The closest point on z1 can be 2 units away from z2.
Thus, the minimum value of |z1 - z2| is 2, making the correct answer option 'c'.
To find the minimum value of |z1 - z2| given the conditions on z1 and z2, we need to analyze the geometric representation of these complex numbers.
Conditions Given
- |z1| = 12: This means z1 lies on a circle centered at the origin (0, 0) with a radius of 12.
- |z2 - 3 - 4i| = 5: This indicates that z2 lies on a circle centered at (3, 4) with a radius of 5.
Geometric Interpretation
- The first circle (for z1) is centered at (0, 0) with a radius of 12.
- The second circle (for z2) is centered at (3, 4) with a radius of 5.
Finding the Minimum Distance
1. Distance Between Centers:
- Calculate the distance between the centers of the two circles:
- Distance = sqrt((3 - 0)² + (4 - 0)²) = sqrt(9 + 16) = sqrt(25) = 5.
2. Minimum Distance Calculation:
- The minimum distance between any point on circle z1 and any point on circle z2 occurs when we move from the edge of z1 directly towards the edge of z2.
- This is calculated as: Distance between centers - radius of z2 - radius of z1.
- Minimum Distance = 5 - 5 - 12 = -12 (not valid).
- However, we need to consider the edges of the circles. The effective distance we need is:
- Minimum Distance = Distance between centers + radius of z1 - radius of z2 = 5 + 12 - 5 = 12.
3. Minimum Value of |z1 - z2|:
- The closest point on z1 can be 2 units away from z2.
Thus, the minimum value of |z1 - z2| is 2, making the correct answer option 'c'.
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For all complex numbers z1 ,z2 satisfying |z1| = 12 and |z2 - 3 - 4i| = 5 , the minimum value of|z1 -z2|isa)0b)7c)2d)17Correct answer is option 'C'. Can you explain this answer?
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