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Question

If z1 and z2 are two roots of the equation z2+az+b=0, z being a complex number. Further assume that the origin, z1 and z2 form an equilateral triangle. Then


A

a2=b

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B

a2=2b

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C

a2=3b

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D

a2=4b

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Solution

The correct option is C

a2=3b


Explanation for the correct option.

Find the relation between a and b.

It is given that the roots of the equation z2+az+b=0 are z1 and z2.

So, the sum of roots is -a, thus z1+z2=-a.

And the product of roots is b, so z1z2=b.

Now, it is given that the origin, z1 and z2 form an equilateral triangle. So, z2=z1eiπ3.

Now, using eiθ=cosθ+isinθ it can be written as:

z2=z1eiπ3z2=z1cosπ3+isinπ3z2=z112+i32cosπ3=12;sinπ3=322z2=z1+i3z12z2-z1=i3z1

Now, square both sides.

2z2-z12=i3z124z22-4z2z1+z12=i23z124z22-4z2z1+z12=-3z12i2=-14z22-4z2z1+z12+3z12=04z12+z22-z1z2=0z12+z22=z1z2

Now, add 2z1z2 both sides.

z12+z22+2z1z2=z1z2+2z1z2z1+z22=3z1z2a+b2=a2+b2+2aba2=3bz1+z2=a,z1z2=b

So, the relation between a and b for the given condition is a2=3b.

Hence, the correct option is C.


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