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प्रश्न
If the real part of `(barz + 2)/(barz - 1)` is 4, then show that the locus of the point representing z in the complex plane is a circle.
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उत्तर
Let z = x + iy
∴ `barz` = x – iy
So`(barz + 2)/(barz - 1) = (x - iy + 2)/(x - iy - 1)`
= `((x + 2) - iy)/((x - 1) - iy)`
= `((x + 2) - iy)/((x - 1) - iy) xx ((x - 1) + iy)/((x - 1) + iy)`
= `((x + 2)(x - 1) + (x + 2)yi - (x - 1)yi - i^2y^2)/((x - 1)^2 - i^2y^2)`
= `(x^2 + 2x - x - 2 + (x + 2 - x + 1)yi + y^2)/((x - 1)^2 + y^2)`
= `(x^2 + y^2 + x - 2)/((x - 1)^2 + y^2) + (3y)/((x - 1)^2 + y^2)i`
Real part = 4
∴ `(x^2 + y^2 + x - 2)/((x - 1)^2 + y^2)` = 4
⇒ x2 + y2 + x – 2 = 4[(x – 1)2 + y2]
⇒ x2 + y2 + x – 2 = 4[x2 + 1 – 2x + y2]
⇒ x2 + y2 + x – 2 = 4x2 + 4 – 8x + 4y2
⇒ x2 – 4x2 + y2 – 4y2 + x + 8x – 2 – 4 = 0
⇒ – 3x2 – 3y2 + 9x – 6 = 0
⇒ x2 + y2 – 3x + 2 = 0
Which represents a circle.
Hence, z lies on a circle.
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