Question

Prove that locus of z is circle and find its centre and radius if is purely imaginary.

Answer 1

Now, `("z" - "i")/("z"-1) = ("x" + "iy" - "i")/("x" + "iy" - 1)` 

 

`= ("x" + "i"("y" - 1))/(("x" - 1)+"iy")`       (∵ z = x + iy)

Rationalising

`= ({"x" + "i" ("y"-1)}{("x"-1) - "iy"})/({("x" -1) + "iy"}{("x"-1)-"iy"})`

 

`= ("x"("x" - 1) + "y"("y"-1)+"i" ("y"-1)("x" - 1)-"xy")/(("x" - 1)^2 + "y"^2)`

 

`=> ("x"^2 + "y"^2 - "x" - "y")/("x"^2 + "y"^2 - "2x" + 1) = 0` , (Real part)          ....[∵ it is purely imaginary]

⇒ x² + y² - x - y = 0

Which is a circle with centre `(1/2 , 1/2)` i.e. `1/2 (1+"i")`  and radius is

`"r" = sqrt((1/2) + (1/2)^2 - 0)` 

 

`= sqrt (1/4 + 1/4) = sqrt(1/2) = 1/sqrt 2`