If z ≠ 1 and z^2/z-1 is real, then the point represented by the complex number z lies: - Mathematics MCQ AIEEE 2012 2026 | ExamTest.live

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If z \neq = 1 and \frac{z ^{2}}{z - 1} is real, then the point represented by the complex number z lies:

Solution & Step-by-step Explanation

Let z = x + i y . \frac{z ^{2}}{z - 1} = \frac{( x + i y ) ^{2}}{( x - 1 ) + i y} = \frac{( x ^{2} - y ^{2} ) + i ( 2 x y )}{( x - 1 ) + i y} Multiply numerator and denominator by (x - 1) - i y : \frac{[( x ^{2} - y ^{2} ) + i ( 2 x y )] [( x - 1 ) - i y ]}{( x - 1 ) ^{2} + y ^{2}} For this to be real, the imaginary part must be zero: - (x^{2} - y^{2}) y + 2 x y (x - 1) = 0 y [- (x^{2} - y^{2}) + 2 x^{2} - 2 x] = 0 y [x^{2} + y^{2} - 2 x] = 0 This gives y = 0 (the real axis) or x^{2} + y^{2} - 2 x = 0 .The equation x^{2} + y^{2} - 2 x = 0 represents a circle with centre (1, 0) and radius 1 . This circle passes through the origin (0, 0) .

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If z \neq = 1 and \frac{z ^{2}}{z - 1} is real, then the point represented by the complex number z lies:

either on the real axis or on a circle passing through the origin

on a circle with centre at the origin

either on the real axis or on a circle not passing through the origin

on the imaginary axis

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