+85 Let z be a complex number such that \(|z - 12| + |z - 5i| = 13.\) Find the smallest possible value of z. Thanks!
+1768 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.
