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AP EAMCET201822 Apr 2018Evening ShiftMathematicsComplex NumberActual

If a point P denotes a complex number z=x+i y in the argand plane and if z+1 z+i is a purely real number, then the locus of P is

Options

  1. Ax+y+1=0
  2. Bx^2+y^2+x+y=0
  3. Cx^2+y^2+2 y+1=0,(x, y) (0,-1)
  4. Dx+y+1=0,(x, y) (0,-1)

Correct answer

D. x+y+1=0,(x, y) (0,-1)

Step-by-step solution

Since aligned & z+1 z+i = x+i y+1 x+i y+i = (x+1) x+i(y+1) & x-i(y+1) x-i(y+1) aligned If z+1 z+i is a purely real number, z -i aligned & Then, I_m ( z+1 z+i )=0, z=-i & x y-(x+1)(y+1)=0, (x, y) (0,-1) & x y-x y-x-y-1=0 & x+y+1=0, (x, y) (0,-1) aligned

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