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Question Number 83127 by jagoll last updated on 28/Feb/20

what is the range of x \sqrt{3} + y

if x^{2} + y^{2} - xy = 3 ?

Answered by mr W last updated on 28/Feb/20

l e t u = \sqrt{3} x + y

y = u - \sqrt{3} x

x^{2} + {(u - \sqrt{3} x)}^{2} - x (u - \sqrt{3} x) = 3

(4 + \sqrt{3}) x^{2} - (1 + 2 \sqrt{3}) u x + u^{2} - 3 = 0

s u c h t h a t x \in R,

Δ = {(1 + 2 \sqrt{3})}^{2} u^{2} - 4 (4 + \sqrt{3}) (u^{2} - 3) ⩾ 0

- 3 u^{2} + 12 (4 + \sqrt{3}) ⩾ 0

u^{2} ⩽ 4 (4 + \sqrt{3})

- 2 \sqrt{4 + \sqrt{3}} ⩽ u ⩽ 2 \sqrt{4 + \sqrt{3}}

i . e . t h e r a n g e o f x \sqrt{3} + y i s [- 2 \sqrt{4 + \sqrt{3}}, 2 \sqrt{4 + \sqrt{3}}]

Commented by jagoll last updated on 28/Feb/20

thank you mister

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