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Question

z1, z2, z3 are vertices of a triangle
Match the following functions given in Column-I with the ranges given in Column-ll

Column-I Column-II
1. z12+z22+z32=z2z3+z3z1+z1z2 (p) right angled
2. Re(z3z1z3z2)=0 (q) obtuse angled
3. Re(z3z1z3z2)<0 (r) Isosceles and right angled
4. z3z1z3z2=i (s) equilateral

A
1s
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B
2p
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C
3q
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D
4r
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Solution

The correct options are
A 2p
B 3q
C 1s
D 4r
We know that
z21+z22+z23=z1z2+z2z3+z3z1 holds for equilateral triangle.
Now
Let ABCD be a polygon,
A=z1
B=z2
C=z3
D=z4
Now Re(z3z1z3z2)=0 indicates that it is purely real. That is its argument is π2
Hence right angled.
Now
consider Re(z3z1z3z2)<0
That is cosθ<0
This is possible if θ is obtuse and does not lie in the fourth quadrant.
If
z3z1z3z2=i
Then Re(z3z1z3z2)=0 indicates that it is purely real. That is its argument is π2
Hence right angled, and
|z3z1|=|z3z2|
Hence isosceles.
All options are correct.

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