Argument of random complex number lying in a given line segment .

Ricky_15

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Q) If z = -5 - 3i & z' = -3 + 5i and z" lies on the line segment joining z & z' , then find the suitable answer from the following options .

A) ## -3\pi/4 ##

B) ## \pi/4 ##

C) ## 5\pi/6 ##

D) ## \pi/6 ##

My solving so far :

Drawing the rough diagram we get to know that x either lies on x-axis or 2nd or 3rd quadrant .
But since there is no option ##\pi## => x lies in either 2nd or 3rd quadrant in the argand plane . => one of ##-3\pi/4 or 5\pi/6 ## the suitable solutions .

I don't know how to proceed further and elimate one of the above two , so as to find the answer to the question .

Book say's : ## 5\pi/6 ## is the correct answer .


 
Q) If z = -5 - 3i & z' = -3 + 5i and z" lies on the line segment joining z & z' , then find the suitable answer from the following options .
What in the world does the notation \(\displaystyle \large \#\#~\theta~\#\#\) mean?
In thirty years of teaching complex variables, I have never seen that used.
 
Q) If z = -5 - 3i & z' = -3 + 5i and z" lies on the line segment joining z & z' , then find the suitable answer from the following options .

A) ## -3\pi/4 ##

B) ## \pi/4 ##

C) ## 5\pi/6 ##

D) ## \pi/6 ##

My solving so far :

Drawing the rough diagram we get to know that x either lies on x-axis or 2nd or 3rd quadrant .
But since there is no option ##\pi## => x lies in either 2nd or 3rd quadrant in the argand plane . => one of ##-3\pi/4 or 5\pi/6 ## the suitable solutions .

I don't know how to proceed further and elimate one of the above two , so as to find the answer to the question .

Book say's : ## 5\pi/6 ## is the correct answer .


The argument of z = (x,y) = x + i y is tan-1(y/x) in the appropriate quadrant, i.e. what is referred to sometimes as atan2(y,x). Since you have eliminated all but \(\displaystyle -\frac{3\, \pi}{4}\, and\, \frac{5\, \pi}{6}\), what would be the relationship of x and y if one of those were the argument of z''? What is the relationship of y and x if z'' lies on the line between z and z'?
 
Q) If z = -5 - 3i & z' = -3 + 5i and z" lies on the line segment joining z & z' , then find the suitable answer from the following options .
A) ## -3\pi/4 ##
B) ## \pi/4 ##
C) ## 5\pi/6 ##
D) ## \pi/6 ##
Book say's : ## 5\pi/6 ## is the correct answer .
I am, glad to see we have a mind reader here.
The answer you need is a number between \(\displaystyle \pi-\arctan\left( \frac{3}{5}\right)~\&~\pi+\arctan \left(\frac{5}{3}\right)\)
 
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