Question

What is a.when z+a/z-a is a purely imaginary number and |z|=82?

Answer 1

a = 82 when z+a/z-a is a purely imaginary number and |z|=82

Step-by-step explanation:

Let say z = x + iy

|z| = 82  => √x² + y²  = 82

z = x + iy

(z + a) / (z - a)

= ( x + iy + a) / ( x + iy - a)

= (x + a + iy) /(x - a + iy)

= ((x + a) + iy) /((x - a) + iy)   * ((x - a) - iy) /((x - a) - iy)

= (x² - a² + y²  + iy(x - a - x - a)) / ( x-a)² + y²)

= (x² - a² + y² )/  ( x-a)² + y²)   - i2ay/ ( x-a)² + y²)

as number is purely imaginary

Hence (x² - a² + y² )/  ( x-a)² + y²)   = 0

=> x² - a² + y²  = 0

=> a² = x² + y²

=> a = √x² + y²

=> a = 82

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