Q. If `z=x+iy` satisfies the equation `arg(z-2) = arg(2z+3i)` then `3x-4y` is equal to:- A) 5 B) -3 C) 6 D) 7

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The complex number z = x + i*y. It is given that arg(z - 2) = arg(2z + 3i).

arg(z - 2) = `tan^-1(y/(x-2))`

arg(2z + 3i) = `tan^-1((2y+3)/(2x))`

`tan^-1(y/(x-2))` = `tan^-1((2y+3)/(2x))`

=> `y/(x - 2) = (2y + 3)/(2x)`

=> 2xy = (2y+3)(x-2)

=> 2xy = 2xy+3x-4y-6

=> 3x - 4y = 6

The correct answer is option C.

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