If z 1 and z 2 be two non zero complex numbers such that z 1- z 2- z 2- z 1-1- then the origin and the points represented by z 1 and z 2A- lie on a straight lineB- form a right angled triangleC- form an isosceles triangleD- form an equilateral triangle

The correct option is D form an equilateral trianglez_1z_2 + z_2z_1 = 1 ⇒ z_1^2+z_2^2=z_1z_2 ...(1) If z_1,z_2,z_3 are the vertices of an equilateral triang ...

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Standard XII

Mathematics

Vertex

If z 1 and z ...

Question

If $z_{1}$ and $z_{2}$ be two non zero complex numbers such that $\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}} = 1,$ then the origin and the points represented by $z_{1}$ and $z_{2}$

lie on a straight line

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form a right angled triangle

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form an equilateral triangle

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form an isosceles triangle

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Solution

The correct option is C form an equilateral triangle
$\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}} = 1$
$\Rightarrow z_{1}^{2} + z_{2}^{2} = z_{1} z_{2} . . . (1)$

If $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle, then
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$
When $z_{3}$ is origin i.e.,
$z_{3} = 0$
$\Rightarrow z_{1}^{2} + z_{2}^{2} = z_{1} z_{2}$ ,
This is same as eqn(1).

Hence, the origin and the points represented by $z_{1}$ and $z_{2}$ form an equilateral triangle.

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If z1 and z2 be two non zero complex numbers such that z1z2+z2z1=1, then the origin and the points represented by z1 and z2

If $z_{1}$ and $z_{2}$ be two non zero complex numbers such that $\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}} = 1,$ then the origin and the points represented by $z_{1}$ and $z_{2}$