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Modulus of a Complex Number

If z1 = 1+i...

Question

If $z_{1} = 1 + i$ and $z_{2} = 1 - i$ are the two vertices of an equilateral triangle then third vertex is given by

\pm \sqrt{3}

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1 \pm \sqrt{3}

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\sqrt{3} \pm 1

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None of these

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Solution

The correct option is A $1 \pm \sqrt{3}$
Let the third vertex be $z_{3}$
Hence all the three vertices must satisfy
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$
$(1 + i)^{2} + (1 - i)^{2} + z_{3}^{2} = (1 - i) (1 + i) + (1 - i) z_{3} + (1 + i) z_{3}$
$1 - 1 + 2 i + 1 - 1 - 2 i + z^{2} = 1 - (- 1) + 2 z$
$z^{2} - 2 z - 2 = 0$
$z = \frac{2 \pm \sqrt{4 + 8}}{2}$
$= 1 \pm \sqrt{3}$
Hence, option 'B' is correct.

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If z1=1+i and z2=1−i are the two vertices of an equilateral triangle then third vertex is given by

The correct option is A 1±√3Let the third vertex be z3Hence all the three vertices must satisfyz21+z22+z23=z1z2+z2z3+z3z1(1+i)2+(1−i)2+z23=(1−i)(1+i)+(1−i)z3+(1+i)z31−1+2i+1−1−2i+z2=1−(−1)+2zz2−2z−2=0z=2±√4+82=1±√3Hence, option 'B' is correct.

If $z_{1} = 1 + i$ and $z_{2} = 1 - i$ are the two vertices of an equilateral triangle then third vertex is given by