Let z be a complex number such that |z+2|=1 and lm((z+1)/(z+2))=1/5. Then the value of |Re(z̅+̅2̅)| is

Solution

Maths sir · 4 steps

Call z+2 = w. It sits on the unit circle, so 1/w is just w̅.

1. Substitute. Let w = z+2, so |w| = 1 and z+1 = w-1. The fraction becomes (w-1)/w = 1 - 1/w.
2. Use the unit circle. Since ww̅ = |w|² = 1, we get 1/w = w̅. Write w = cos θ + i sin θ, so w̅ = cos θ - i sin θ.

   Im(1 - w̅) = sin θ = 1/5

3. Find the real part. cos² θ = 1 - 1/25 = 24/25, so cos θ = ±√24/5 = ±2√6/5. Conjugation does not change the real part, so Re(z̅+̅2̅) = cos θ.
4. So. |Re(z̅+̅2̅)| = 2√6/5, which is option (a).

The idea to keep. Whenever |w| = 1, replace 1/w by w̅ and write w = cos θ + i sin θ. A messy quotient usually turns into one line of trigonometry.

Answer from the NTA final answer key. Solution by PrepWiser.