if-x-2-y-2-xy-5-find-the-range-of-x-2-y-2-xy-x-y-R-

Question Number 179100 by mr W last updated on 24/Oct/22

i f x^{2} + y^{2} + x y = 5, f i n d t h e r a n g e o f

x^{2} + y^{2} - x y .

(x, y \in R)

Commented by Frix last updated on 24/Oct/22

[\frac{5}{3}; 15]

Commented by Acem last updated on 25/Oct/22

G o o d!

Answered by Acem last updated on 25/Oct/22

R = [- 5, + \infty [

Commented by MJS_new last updated on 25/Oct/22

how ?

Answered by MJS_new last updated on 25/Oct/22

let y = p x

(p^{2} + p + 1) x^{2} = 5

x^{2} = \frac{5}{p^{2} + p + 1}

x^{2} + y^{2} - x y = \frac{5 (p^{2} - p + 1)}{p^{2} + p + 1}

\frac{d [\frac{5 (p^{2} - p + 1)}{p^{2} + p + 1}]}{d p} = 0

\frac{10 (p^{2} - 1)}{{(p^{2} + p + 1)}^{2}} = 0 \Rightarrow p = \pm 1

\Rightarrow

\frac{5}{3} ⩽ \frac{5 (p^{2} - p + 1)}{p^{2} + p + 1} ⩽ 15

\Leftrightarrow

\frac{5}{3} ⩽ x^{2} + y^{2} - x y ⩽ 15

Commented by Acem last updated on 25/Oct/22

{Y o u}^{'} r e r i g h t

Commented by mr W last updated on 25/Oct/22

y e s, t h a n k s!

Answered by mr W last updated on 25/Oct/22

x^{2} + y^{2} + x y = 5 \dots (i)

x^{2} + y^{2} - x y = k \dots (i i)

(i) + (i i) :

x^{2} + y^{2} = \frac{5 + k}{2}

(i) - (i i) :

x y = \frac{5 - k}{2}

\frac{5 - k}{2} = x y ⩽ \frac{x^{2} + y^{2}}{2} = \frac{5 + k}{4}

\Rightarrow 10 - 2 k ⩽ 5 + k \Rightarrow k ⩾ \frac{5}{3}

\frac{5 - k}{2} = x y ⩾ - \frac{x^{2} + y^{2}}{2} = - \frac{5 + k}{4}

\Rightarrow 10 - 2 k ⩾ - 5 - k \Rightarrow k ⩽ 15

\Rightarrow \frac{5}{3} ⩽ x^{2} + y^{2} - x y ⩽ 15

Commented by mr W last updated on 25/Oct/22

{(x - y)}^{2} ⩾ 0 \Rightarrow x^{2} + y^{2} ⩾ 2 x y \Rightarrow x y ⩽ \frac{x^{2} + y^{2}}{2}

{(x + y)}^{2} ⩾ 0 \Rightarrow 2 x y ⩾ - (x^{2} + y^{2}) \Rightarrow x y ⩾ - \frac{x^{2} + y^{2}}{2}

Commented by Acem last updated on 25/Oct/22

N i c e s o l u t i o n, b u t

\frac{5 - k}{2} = x y ⩽ \frac{5 + k}{2} b u t h o w x y ⩽ \frac{5 + k}{4}

Commented by mr W last updated on 25/Oct/22

x y ⩽ \frac{x^{2} + y^{2}}{2} a n d x^{2} + y^{2} = \frac{5 + k}{2}, s o i t i s

c l e a r t o m e t h a t

x y ⩽ \frac{5 + k}{4} .

Commented by Acem last updated on 26/Oct/22

G o o d

Answered by manxsol last updated on 26/Oct/22

f i n d r^{2} (1 - \frac{s i n 2 θ}{2})

r^{2} (1 + \frac{s i n 2 θ}{2}) = 5

s i n 2 θ = \frac{10}{r^{2}} - 2

- 1 ≪ \frac{10}{r^{2}} - 2 ⩽ 1

\frac{10}{3} ⩽ r^{2} ⩽ 10

\frac{1}{2} ⩽ 1 - \frac{s i n 2 θ}{2} ⩽ \frac{3}{2}

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