If - z -1 and z 1- z 3- z 2- z 3- z - i - z - i then z 1- z 2- z 3 will be vertices of aA- None of theseB- equilateral triangleC- acute angled triangleD- obtuse angled triangle

As | z |=1 lies on circle, so I and I are extremities of this circle. So, arg (z i/z+i) = ±π/2 ⇒ z i/z+i is purely imaginary ⇒ z_1, z_2, z_3 represents verti ...

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Byju's Answer

Standard XII

Mathematics

Properties of Modulus

If | z |=1 an...

Question

If $| z | = 1$ and $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{z - i}{z + i}$ then $z_{1}, z_{2}, z_{3}$ will be vertices of a

equilateral triangle

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acute angled triangle

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obtuse angled triangle

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None of these

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Solution

The correct option is D None of these
As $| z | = 1$ lies on circle, so I and -I are extremities of this circle.

So, arg $(\frac{z - i}{z + i})$ = $\pm \frac{π}{2}$ $\Rightarrow$ $\frac{z - i}{z + i}$ is purely imaginary
$\Rightarrow z_{1}, z_{2}, z_{3}$ represents vertices of right angled triangle

Suggest Corrections

Similar questions

Q. If

| z | = 1

and

\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{z - i}{z + i}

then

z_{1}, z_{2}, z_{3}

will be vertices of a

Q. If

| z | = 1

and

\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{z - i}{z + i}

then

z_{1}, z_{2}, z_{3}

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Q. If

| z | = 2

and

\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{z - 2}{z + 2}

Then show that

z_{1}, z_{2}, z_{3}

are vertices of a right-angled triangle.

z_{1}, z_{2}, z_{3}

are vertices of equilateral triangle inscribed in the circle

| z | = 2

z_{1} = 1 + i \sqrt{3}

then

z_{2}

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z_{3}

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Q. Assertion :Let

z_{1}, z_{2}, z_{3}

be distinct complex numbers &

ω^{3} = 1, ω \neq 1

z + ω z_{2} + ω^{2} z_{3} = 0

then

z_{1}, z_{2}, z_{3}

are the vertices of an equilateral triangle. Reason: If

z_{3} - z_{1} = (z_{2} - z_{1}) e^{- 1^{π} / 3}

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If |z|=1 and z1− z3z2− z3= z−iz+i then z1, z2, z3 will be vertices of a

The correct option is D None of theseAs |z|=1 lies on circle, so I and -I are extremities of this circle. So, arg (z−iz+i) = ±π2 ⇒ z−iz+i is purely imaginary ⇒z1,z2,z3 represents vertices of right angled triangle

If $| z | = 1$ and $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{z - i}{z + i}$ then $z_{1}, z_{2}, z_{3}$ will be vertices of a