Beautiful problem
Four distinct points z , z 2 , z 3 , and z 4 in the Argand plane lie on a circle.
Find ∣ z ∣ .
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3 solutions
This was NICE!!!! An interesting method!!
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@Vilakshan Gupta Tera bhi yhi method thaa kya??? Yaa fir koi aur??
I didn't know about the condition of concyclicity, so couldn't solve it.
But my sir gave an elegant solution to it, and that's why the name I gave to the problem.
I am writing the solution, you see it.
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Oooh okay......I'm waiting!! :P
Btw, check my new problem.....!!
You can see it now!
z = e j θ ∣ z N ∣ = 1
Therefore, z raised to any integer power lies on the unit circle.
What is j ?
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j represents the imaginary unit, − 1 . Most commonly it's represented by i .
@Steven Chase Sir, there's an error......the argument of the complex number z can only belong to the intervals ( 0 , 3 2 π ) or ( 3 4 π , 2 π )
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The "large" formatting I originally used doesn't work anymore. I have fixed the formatting. Do you still have a concern about it?
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Sir, I think you didn't get my point. The only z which satisfy the given condition are those which lie on the unit circle and have their polar angles belonging to the ranges stated above, otherwise the four points won't be distinct anymore....
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@Aaghaz Mahajan – Ok, that's fine. My intent was only to comment on the necessity of the magnitude being one.
Assume that the centre of the circle on which they lie is z 0 .
Now multiplying all points by z just shifts the circle by an argument of z and the circle is rescaled according to the magnitude of z .
But here note that since we multiplied by the same number, the points now will become z 2 , z 3 , z 4 , z 5 and the centre will become z z 0 .
Here observe that the points z 2 , z 3 , z 4 are common to both the circles i.e (shifted one and original one) , so what can be concluded?
We conclude that the circle is the same circle i.e it didn't change! (Why? Because 3 points lie only on one circle)
So that means the centre is same, that is, z z 0 = z 0 which implies either z 0 = 0 or ∣ z ∣ = 1 , but the formal condition itself means ∣ z ∣ = 1 because it would be a circle centred at origin on which the given points lie , so their modulus has to be 1 .