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Question

If z1 and z2z1 is real, then the point represented by the complex number z lies:
  1. either on the real axis or on a circle passing through the origin.
  2. on a circle with centre at the origin.
  3. either on the real axis or on a circle not passing through the origin.
  4. on the imaginary axis

A
either on the real axis or on a circle not passing through the origin.
B
either on the real axis or on a circle passing through the origin.
C
on the imaginary axis
D
on a circle with centre at the origin.
Solution
Verified by Toppr

z1 and z2z1 is real so imaginary Part is 0
Lets say z=x+iy
So expression =(x+iy)2x+iy1=x2y2+2ixy(x1)+iy
Rationalisation x2y2+2ixy(x1)+iy×(x1)iy(x1)iy
=(x2y2+2ixy)((x1)iy)(x1)2+y2
(Imaginary Part =0)2xy(x1)x2y+y3=0
2x2y2xyx2y+y3=0
x2y+y32xy=0
y(x2+y22x)=0
y=0 or x2+y22x=0
Real axis Line:y=0orCircle:x2+y22x=0(x1)2+(y0)2=12

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