I have a maths. question from the Class XII Mathematics book by R.D.Sharma ( Chapter Complex numbers ) . I request anyone who can solve the problem step by step and send me the same The problem is as under : Locate the region represented by the complex number z in the Argand plane in case of the following condition : log((pi/6),((|z-2|+5)/(4*|z-2|-4)))<2. Please send step by step solution for the above question. Regards, S.B.Roy

I reckon that Mr. Bhushan did a typo error and the original problem is something like this, Located the region represented by the complex number z in the Argand plane in case of the following condition : log(pi/6) < (|z-2| + 5) / (4*|z-2| - 4) < 2 The detailed solution as requested by you is as follows: |z-2| + 5 log(pi/6) < --------------- < 2 4|z-2| - 4 |z-2| + 5 therefore 4 * log(pi/6) < ---------------------- < 8 |z-2| - 1 Writing 5 = -1+6, we get (|z-2| -1) + 6 4log(pi/6) < -------------------------- < 8 ( |z-2| -1 ) 6 4 log(pi/6) < 1 + --------------- < 8 |z-2| -1 6 4log(pi/6) - 1 < ------------------ < 7 |z-2| - 1 INVERTING THE INEQUALITY GIVES US 1 |z-2| - 1 1 ------- < ---------------- < ----------------------- 7 6 4log(pi/6) -1 6 6 ---------- < |z-2| -1 < ------------------------ 7 4log(pi/6) - 1 13 6 --------- < |z-2| < -------------------------- + 1 7 4log(pi/6) - 1 13 6 + 4log(pi/6) -1 -------- < |z-2| < ------------------------------ 7 4log(pi/6) - 1 But |z-2| = | x + yi -2| = | ( x - 2 ) + yi | = ( x - 2) ^2 + y ^2 is the equation of a circle, hence our problem becomes 13 5 + 4log(pi/6) -------- < ( x - 2) ^2 + y ^2 < ------------------------------ 7 4log(pi/6) - 1 Or, A < ( x - 2) ^2 + y ^2 < B with A and B corresponding to the above values i.e A = 13 / 7 and B = the complicated fraction on the right The above inequality can be interpreted as the region between two concentric circles centered at (2,0) with two different radii. Smaller circle with radius = square root (A) and centered at (2,0) and larger circle with radius = square root (B) and centered at (2,0). Your solution is the entire region between these two circles. My only concern is that log(pi/6) is essentially negative and I have no clue as to which base are you using to calculate log(pi/6). This has serious consequences if you know what I mean. I hope this is clear now. May I suggest Complex Variables by Churchill and Brown as a good book to learn more about complex variables. Cheers read less