Let complex number z=(a+cos θ)+(2a−sin θ)i . If ∣z∣ ≤2 for any θ∈R then the range of real number a is __


Question and Answers Forum

All Questions Topic List
Number Theory Questions
Previous in All Question Next in All Question
Previous in Number Theory Next in Number Theory
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

	$nice prof . .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum