Suppose z=a+bi, where a and b are integers and i is the imaginary unit. We are given that |1+iz|=|1−iz| and |z−(13+15i)|<17. Find the largest possible value of a+b.
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To find the largest possible value of a + b, we first need to analyze the given conditions and derive the necessary equations.
Let's start by simplifying the first condition, |1 + iz| = |1 - iz|. Using the definition of modulus, we have:
|1 + iz| = |1 - iz|
√((1 + iz)(1 + iz)*) = √((1 - iz)(1 - iz)*)
√(1 + 2iz - z²) = √(1 - 2iz - z²)
Squaring both sides of the equation, we get:
1 + 2iz - z² = 1 - 2iz - z²
The imaginary terms cancel out, leaving us with:
4iz = 0
Since z = a + bi, where a and b are integers, the imaginary part of z (bi) can only equal zero for z to be real. Thus, this condition simplifies to:
b = 0
Now, let's analyze the second condition, |z - (13 + 15i)| < 17. Using the distance formula (also known as the modulus formula), we have:
|(a + bi) - (13 + 15i)| < 17
|(a - 13) + (b - 15)i| < 17
We can treat this as a distance problem in the complex plane. The inequality represents all the complex numbers whose distance from (a - 13) + (b - 15)i is less than 17. This forms a circle with radius 17 centered at (13, 15) in the complex plane.
Now, to find the largest possible value of a + b, we need to find the point (a, b) on or within this circle that maximizes a + b.
Since we know that b = 0, our search narrows to the x-axis. To find the point (a, 0) on or within the circle, we need to find the x-coordinate (a) that is at a maximum distance of 17 from (13, 15) on the x-axis.
The x-coordinate in this case will be either 13 + 17 or 13 - 17. Taking the larger value gives us:
a = 13 + 17
a = 30
Finally, adding the values of a and b:
a + b = 30 + 0
a + b = 30
Therefore, the largest possible value of a + b is 30.