IIT-JEE 2007 · Paper 2 · Mathematics Q46

MathematicsSingle correct+3 / -1Question 46 of 66

If $|z|=1$ and $z\ne\pm1$ , then all the values of $\dfrac{z}{1-z^2}$ lie on

a line not passing through the origin

|z|=\sqrt2

the

x

-axis

the

y

-axis

Answer and worked solution

Official answer: D

Use $\bar z=\frac1z$ . For $|z|=1$ ,
$\frac z{1-z^2}=\frac1{\frac1z-z}=\frac1{\bar z-z}=\frac1{-2i\,\mathrm{Im}\,z}=\frac{i}{2\,\mathrm{Im}\,z}.$

Purely imaginary. With $z\ne\pm1$ , $\mathrm{Im}\,z\ne0$ , and the value is a nonzero imaginary number: it lies on the y-axis.

(D)

Topics:Geometry in the complex plane Modulus and argument

IIT-JEE 2007 · Paper 2, Mathematics question 46