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Question

The points (α,β),(γ,δ)(α,δ) and (γ,β) taken in order, where α,β,γ,δ are different real numbers, are

A
collinear
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B
vertices of a square
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C
vertices of a rhombus
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D
concyclic
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Solution

The correct option is D concyclic
The given points are not collinear as we can see they form a rectangle.
The side's length are not equal which means it is not a square or rhombus.
Which only leaves concyclic.
Using Ptolemy's theorem
Product of diagonal=Sum of pair of opposite sides then the quadrilateral ca be inscribed in the circle
Both diagonal's d=(αγ)2+(βδ)2
Rectangle length(l)=(αγ) and breadth(b)=(βδ)
dd=(αγ)2+(βδ)2
ll=(αγ)2 and bb=(βδ)2
dd=ll+bb Therefore they are concyclic.

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