IGNOU MMTE 5 2026 - Find the image of the circle under the mapping .

Question

Find the image of the circle $|z| = r (r \neq 1)$ under the mapping $w = f(z) = \frac{z - i}{z + i}$ . What happens when $r = 1$ ?

09 Jan 2026

Answer :

Word Count : 528

Let ( z = x + iy ), where ( x, y \in \mathbb{R} ). The mapping is [ w = \frac{z - i}{z + i} = \frac{x + i(y-1)}{x + i(y+1)}. ] To find the image, write ( w = u + iv ), where ( u, v \in \mathbb{R} ). Multiply numerator and denominator by the conjugate of the denominator: [ w = \frac{x + i(y-1)}{x + i(y+1)} \cdot \frac{x - i(y+1)}{x - i(y+1)} = \frac{x^2 + (y-1)(y+1) + i[-x(y+1) + x(y-1)]}{x^2 + (y+1)^2}. ] Simplify numerator: * Real part: ( x^2 + y^2 - 1 ) * Imaginary part: ( i[-x(y+1) + x(y-1)] = i[-xy - x + xy - x] = -2ix ) Denominator: ( x^2 + y^2 + 2y + 1 ) So [ w = \frac{(x^2 + y^2 - 1) - 2ix}{x^2 + y^2 + 2y + 1}. ] Given ( |z| = r ), i.e., ( x^2 + y^2 = r^2 ), then [ w = \frac{r^2 - 1 - 2ix}{r^2 + 2y + 1}. ] Let ( u = \Re(w) ), (

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