Let the curve z(1+i)+zˉ(1−i)=4,z∈C, divide the region ∣z−3∣≤1 into two parts of areas α and β. Then ∣α−β∣ equals:
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Step-by-Step Solution
Step 1: Convert the complex equation to Cartesian form
We are given a complex equation for a curve. To understand its geometric representation, we substitute z=x+iy and zˉ=x−iy into the equation. This allows us to convert the complex equation into its equivalent Cartesian form, which will reveal the type of curve.
Step 2: Simplify the equation to find the line
After substituting z and zˉ, we expand the terms and combine the real and imaginary parts. The imaginary parts cancel out, leaving us with a linear equation in x and y. This equation, x−y=2, represents a straight line in the Cartesian plane.
Step 3: Identify the region and its properties
The region ∣z−3∣≤1 represents all complex numbers z whose distance from the point 3 (which corresponds to (3,0) in the Cartesian plane) is less than or equal to 1. This describes a closed disk centered at (3,0) with a radius of 1. The total area of this disk is πr2=π(1)2=π.
Step 4: Determine the relationship between the line and the circle
We need to find how the line x−y−2=0 intersects the circle (x−3)2+y2=1. We calculate the perpendicular distance d from the center of the circle (3,0) to the line. Since d=21≈0.707 is less than the radius r=1, the line intersects the circle, dividing it into two segments.
Step 5: Calculate the area of the smaller segment
The line divides the circle into two segments. The area of one segment (let's call it α) can be calculated using the formula for the area of a circular segment. We use the radius r=1 and the distance d=1/2. This gives us α=4π−21.
Step 6: Calculate the difference in areas
The total area of the disk is π. If one part is α, the other part β is π−α. We then find the absolute difference between α and β. Substituting the value of α, we get the final result.