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Let z be a complex number such that \(|z - 12| + |z - 5i| = 13.\) Find the smallest possible value of z. Thanks!

 Jun 4, 2024
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The equation |z - 12| + |z - 5i| = 13 represents the sum of the distances from the complex number z to two fixed points in the complex plane: 12 (on the real number axis) and 5i (on the imaginary number axis).

 

Here's how to solve for the smallest possible value of ∣z∣ (which represents the distance from the origin):

 

Triangle Inequality: The sum of the lengths of any two sides of a triangle must be greater than the absolute value of the difference of the remaining side length.

 

In this case, consider a triangle formed by points z, 12, and 5i. Applying the triangle inequality:

 

First inequality: |z - 12| + |5i - z| >= |12 - (5i)|

 

Second inequality: |z - 5i| + |12 - z| >= |5i - 12|

 

Since both sides of the original equation |z - 12| + |z - 5i| = 13 are positive, we can rewrite it as:

|z - 12| + |z - 5i| = 13

 

Simplifying the inequalities:

 

First inequality:

 

|12 - (5i)| = |-12 + 5i| = sqrt(12^2 + 5^2) = 13 (using the distance formula)

 

Therefore, |z - 12| + |z - 5i| >= 13 (which is the same as the original equation)

 

Second inequality:

 

|5i - 12| = |-12 - 5i| = sqrt((-12)^2 + (-5)^2) = 13 (using the distance formula)

 

Therefore, |z - 5i| + |12 - z| >= 13

 

Smallest possible value of |z|:

 

Since both inequalities we derived involve |z|, the smallest possible value of |z| will occur when both inequalities become equalities.

 

This means that triangle z - 12 - 5i must be an isosceles triangle with base 12−(−5i)=12+5i and both legs having the same length as ∣z∣.

 

In other words, point z must be equidistant to both 12 and 5i.

 

Finding the midpoint:

 

The midpoint of the line segment connecting 12 and 5i is:

 

Midpoint = [(12 + 0i) + (0 + 5i)] / 2 = (6 + 2.5i)

 

Therefore, for the smallest possible value of ∣z∣, point z must coincide with the midpoint, which is (6+2.5i).

 

Conclusion: The smallest possible value of ∣z∣ is the distance between the origin and the midpoint (6+2.5i), which can be found using the distance formula:

 

|z| = sqrt(6^2 + (2.5)^2) = sqrt(41)

 

Therefore, the smallest possible value of z is 6+2.5i​.

 Jun 4, 2024

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