JEE Main 2014 · 12 Apr · Shift 1 · Mathematics Q62

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MathematicsSingle correct+4 / -1Question 62 of 90

Let $z\ne-i$ be any complex number such that $\dfrac{z-i}{z+i}$ is a purely imaginary number. Then $z+\dfrac1z$ is :

any non-zero real number other than 1.

any non-zero real number.

a purely imaginary number.

Answer and worked solution

Official answer: C

$\dfrac{z-i}{z+i}$ purely imaginary means the segment from $-i$ to $i$ subtends a right angle at $z$ , so $z$ lies on the circle with that diameter: $|z|=1$ . Then $\frac1z=\bar z$ and
$z+\frac1z=z+\bar z=2\,\text{Re}\,z,$
a real number. It is non-zero since $z\ne\pm i$ ( $z=i$ makes the ratio 0).

Topics:Geometry in the complex plane Modulus and argument

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JEE Main 2014 · 12 Apr · Shift 1, Mathematics question 62