Let the line z(1+ 3 i) + z (1- 3 i) = k , where k > 4 is a real constant, divide the region |z-2| 2 into two parts. If the absolute difference between the areas of these two parts is 4 3 + 2 3 , then the value of k is :

Step-by-step solution

Let z = x+iy . The equation of the line is: (x+iy)(1+ 3 i) + (x-iy)(1- 3 i) = k x + 3 ix + iy - 3 y + x - 3 ix - iy - 3 y = k 2x - 2 3 y = k x - 3 y - k 2 = 0 The given region is |z-2| 2 , which represents a circle with center (2,0) and radius R = 2 . Total area of the circle is A = (2)^2 = 4 . Let the areas of the two parts be and with > . We are given - = 4 3 + 2 3 . Since + = 4 , subtracting the two equations gives: 2 = 4 - ( 4 3 + 2 3 ) = 8 3 - 2 3 = 4 3 - 3 The area of a circular segment subtending an angle at