Let O, A, B be three collinear points such that O A ·O B=1 . If O and B represent the complex nu

step 1 In this question it is given that three distinct complex numbers, Z1, Z2 and Z3 are collinear. W

Let O, A, B be three collinear points such that O A ·O B=1 ....

Let $O, A, B$ be three collinear points such that $O A \cdot O B=1 .$ If $O$ and $B$ represent the complex numbers $o$ and $z$, then $A$ represents
(A) $\frac{1}{\bar{z}}$
(B) $\frac{1}{z}$
(C) $\bar{z}$
(D) $z^{2}$

Key Concepts

Inversion in the Complex Plane

Inversion is a transformation that maps a point in the complex plane to another point along the same ray from the origin such that the product of the distances from the origin remains constant (often the square of the radius of the inversion circle). For the unit circle, the image of a point z under inversion is given by a formula that typically incorporates complex conjugation to ensure that the image lies on the same line as the original point.

Complex Conjugation

Complex conjugation is the process of changing the sign of the imaginary part of a complex number. This operation is essential in many formulas in complex analysis, particularly in inversion formulas where the conjugate is used to produce an image that has the correct alignment with respect to the real axis and the center of the inversion.

Collinearity in the Complex Plane

Collinearity in the complex plane means that points represented by complex numbers lie on a straight line. This is often checked by showing that one complex number is a real scalar multiple of another. Such a property is key in problems where points are required to be collinear, for example, in determining the image of a point under inversion with respect to a circle centered at the origin.

Scalar Multiplication in Complex Numbers

Scalar multiplication involves multiplying a complex number by a real number. This operation scales the magnitude of the complex number while leaving its direction unchanged. In many geometric transformations, including inversion, ensuring that a point remains on the same line through the origin is achieved by expressing it as a real multiple of another complex number.

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