SOLUTION: The complex numbers {{{ z[1] }}} and {{{ z[2] }}} are connected by the relation {{{ z[1] = z[2] + 1/z[2] }}} If the point representing {{{ z[2] }}} in the Argand diagram describe

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: The complex numbers {{{ z[1] }}} and {{{ z[2] }}} are connected by the relation {{{ z[1] = z[2] + 1/z[2] }}} If the point representing {{{ z[2] }}} in the Argand diagram describe      Log On


   

Question 1161851: The complex numbers +z%5B1%5D+ and +z%5B2%5D+ are connected by the relation
+z%5B1%5D+=+z%5B2%5D+%2B+1%2Fz%5B2%5D+
If the point representing +z%5B2%5D+ in the Argand diagram describes a circle of radius +a+ and centre at the origin, show that the point representing +z%5B1%5D+ describes the ellipse
+x%5E2%2F%281%2Ba%5E2%29%5E2+%2B+y%5E2%2F%281-a%5E2%29%5E2+=+1%2Fa%5E2+
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
There must be an easier, much more elegant way, but this works:

z%5B2%5D=a%28cos%28theta%29%2Bi%2Asin%28theta%29%29 describes the circle of radius a

z%5B1%5D=z%5B2%5D%2B1%2Fz%5B2%5D






The point representing z%5B1%5D=x%2Bi%2Ay has
or
Then,
x%2F%281%2Ba%5E2%29%22=%22
and
meaning that highlight%28x%5E2%2F%281%2Ba%5E2%29%5E2=%281%2Fa%5E2%29%2Acos%5E2%28theta%29%29
Similarly,
y%2F%281-a%5E2%29%22=%22
and
meaning that highlight%28y%5E2%2F%281-a%5E2%29%5E2=%281%2Fa%5E2%29%2Asin%5E2%28theta%29%29
Adding up the equations highlighted above,

meaning that
highlight%28x%5E2%2F%281%2Ba%5E2%29%5E2%2By%5E2%2F%281-a%5E2%29%5E2=1%2Fa%5E2%29