Let-complex-number-z-a-cos-2a-sin-i-If-z-2-for-any-R-then-the-range-of-real-number-a-is-

Question Number 148951 by EDWIN88 last updated on 01/Aug/21

L e t c o m p l e x n u m b e r z = (a + \cos θ) + (2 a - \sin θ) i .

I f ∣ z ∣ ⩽ 2 f o r a n y θ \in R t h e n t h e

r a n g e o f r e a l n u m b e r a i s___

Answered by iloveisrael last updated on 01/Aug/21

Answered by mr W last updated on 01/Aug/21

{(a + \cos θ)}^{2} + {(2 a - \sin θ)}^{2} ⩽ 2^{2} = 4

5 a^{2} + 2 a (\cos θ - 2 \sin θ) - 3 ⩽ 0

5 a^{2} + 2 k a - 3 ⩽ 0

k = \cos θ - 2 \sin θ = \sqrt{5} \cos (θ + \tan^{- 1} 2)

- \sqrt{5} ⩽ k ⩽ \sqrt{5}

5 a^{2} + 2 \sqrt{5} a - 3 ⩽ 0

\Rightarrow a ⩽ \frac{\sqrt{5}}{5}

5 a^{2} - 2 \sqrt{5} a - 3 ⩽ 0

\Rightarrow a ⩾ - \frac{\sqrt{5}}{5}

\Rightarrow - \frac{\sqrt{5}}{5} ⩽ a ⩽ \frac{\sqrt{5}}{5}

Commented by puissant last updated on 01/Aug/21

nice prof . .