Question

if |z-5i|/|z+5i| = 1, then show that z is a real number

Answer 1

Answer:

Step-by-step explanation: assuming z =x+iy.

Z-5i= x+(y-5)i

Z+5i=x+(y+5)i

|Z-5i| = (x^2+(y-5)^2)^1/2

Similarly of other then rationalize it ,you will get coefficient of iota (I) = 0

Therefore it will be a real number

Answer 2

z+5i

z−5i

​

=1

⇒  

x+iy+5i

x+iy−5i

​

=1

⇒  

x+i(y+5)

x+i(y−5)

​

=1

 

x+i(y+5)

x+i(y−5)

​ ×  

x−i(y+5)

x−i(y+5) =1

⇒

x  

2

+(y+5)  

2

 

x  

2

+i(y−5)x−ix(y+5)+y  

2

−25

​

=1

⇒x  

2

+i(yx−5x−xy−5x)+y  

2

−25=x  

2

+y  

2

+10y+25

⇒+i(−5x−5x)−25=10y+25

⇒−10ix−10y=50

⇒−i(x+iy)=5

⇒x+iy=  

−i

5

​

×  

i

i

​

=5i

Comparing the real and imaginary parts we get

∴x=0,y=5

∴z=0+5i=5i