Let the complex numbers α and 1/ᾱ lie on the circles |z-z₀|²=4 and |z-z₀|²=16 respectively, where z₀=1+i. Then, the value of 100|α|² is __________.

Solution

Maths sir · 4 steps

Expand both circles: the same cross term appears twice, so eliminate it and solve for |α|².

1. Expand the first. Use |z - z₀|² = |z|² - (zz̄₀ + z̄z₀) + |z₀|² with |z₀|² = 2. Write r = |α|² and S = αz̄₀ + ᾱz₀. Then r - S + 2 = 4, so S = r - 2.
2. Expand the second. For β = 1/ᾱ we have β̄ = 1/α and |β|² = 1/r. The cross term is z̄₀/ᾱ + z₀/α = (αz̄₀ + ᾱz₀)/|α|² = S/r. It is the same S as before.

   1/r - S/r + 2 = 16 ⇒ 1 - S = 14r

3. Substitute S. Put S = r - 2 into 1 - S = 14r: 3 - r = 14r, so 15r = 3 and r = 1/5.
4. So. |α|² = 1/5, so 100|α|² is 20.

The common slip: 25. Taking |z₀|² = 1 instead of |1 + i|² = 2 gives |α|² = 1/4 and the answer 25.

The idea to keep. When α and 1/ᾱ both appear, expand |z - z₀|² fully. The cross term for 1/ᾱ is the cross term for α divided by |α|², so you can eliminate it.

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