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