if x^2 +y^2 +xy=5, find the range of x^2 +y^2 −xy. (x,y ∈R)


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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

	$x^{2} + y^{2} + x y = 5 . . . (i)$ $x^{2} + y^{2} - x y = k . . . (i i)$ $(i) + (i i) :$ $x^{2} + y^{2} = \frac{5 + k}{2}$ $(i) - (i i) :$ $x y = \frac{5 - k}{2}$ $\frac{5 - k}{2} = x y ⩽ \frac{x^{2} + y^{2}}{2} = \frac{5 + k}{4}$ $\Rightarrow 10 - 2 k ⩽ 5 + k \Rightarrow k ⩾ \frac{5}{3}$ $\frac{5 - k}{2} = x y ⩾ - \frac{x^{2} + y^{2}}{2} = - \frac{5 + k}{4}$ $\Rightarrow 10 - 2 k ⩾ - 5 - k \Rightarrow k ⩽ 15$ $\Rightarrow \frac{5}{3} ⩽ x^{2} + y^{2} - x y ⩽ 15$
	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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