If argument of z − z 1 z − z 2 is π 4 ,...

Cierra Castillo

Cierra Castillo

Answered

2022-07-12

If argument of z z 1 z z 2 is π 4 , find the locus of z.
z 1 = 2 + 3 i
z 2 = 6 + 9 i
Approach: I tried to solve the equation using diagram, basically plotting the points on the Argand plane. What I got is a circle with center 7 + 4 i and a radius of 26 units. The two complex numbers given lie on this circle, and form a chord. Any point lying on the major arc of this chord satisfies the condition.
How exactly would I represent this as a locus of the point? And is there any other method that I can use that does not involve a diagram?

Answer & Explanation

Jenna Farmer

Jenna Farmer

Expert

2022-07-13Added 17 answers

the angle subtended by the chord z 1 z 2 at the center is 2 π / 4 = π / 2 so the radius is | z 1 z 2 | 2 = 26 the center of the chord is 4 + 3 i you add or subtract 6 + 4 i 2 so that you will get two centers. the two centres, z 1 and z 2 form a square of side 26 .
Savanah Boone

Savanah Boone

Expert

2022-07-14Added 5 answers

Put z = x + i y , x , y R , so
z 2 3 i z 6 9 i = ( x 2 ) + ( y 3 ) i ( x 6 ) + ( y 9 ) i ( x 6 ) ( y 9 ) i ( x 6 ) ( y 9 ) i =
= ( x 2 ) ( x 6 ) + ( y 3 ) ( y 9 ) ( x 6 ) 2 + ( y 9 ) 2 + ( x 6 ) ( y 3 ) ( x 2 ) ( y 9 ) ( x 6 ) 2 + ( y 9 ) 2 i
By the given data, it must be that the real and imaginary parts are identical, and thus
( x 2 ) ( x 6 ) + ( y 3 ) ( y 9 ) = ( x 6 ) ( y 3 ) ( x 2 ) ( y 9 )
x 2 14 x + y 2 8 y 26 = 0
Complete squares, make some algebraic hokus pokus and get a circle.

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