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AP EAMCET202420 May 2024Morning ShiftMathematicsComplex NumberActual

If the point p represents the comple x number z=x+i y in the argand plane and if z+i z-1 is a purely imaginary number then the locus of p is

Options

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

Correct answer

B. x^2+y^2-x+y=0 and (x, y) (1,0)

Step-by-step solution

z=x+i y, p=(x, y) z+i z-1 = x+i y+i x+i y-1 = x+i(y+1) (x-1)+i y (x-1)-i y (x-1)-i y = x(x-1)+i(x-1)(y+1)-i x y+y(y+1) (x-1)^2+y^2 = x^2+y^2-x+y (x-1)^2+y^2 + i(x-y+1) (x-1)^2+y^2 Since z+i z-1 is purely imaginary number Re ( z+i z-1 )=0 x^2+y^2-x+y (x-1)^2+y^2 =0 x^2+y^2-x+y=0 and (x, y) (1,0)

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