If-x-and-y-are-real-numbers-then-is-it-possible-that-sec-2xy-x-2-y-2-

Question Number 206160 by MATHEMATICSAM last updated on 08/Apr/24

If x and y are real numbers then is it

possible that \sec θ = \frac{2 x y}{x^{2} + y^{2}} ?

Commented by mr W last updated on 08/Apr/24

o n l y i f x = \pm y .

Answered by mahdipoor last updated on 08/Apr/24

s e c a = \frac{2 x y}{x^{2} + y^{2}} \in] - 1, 1 [\Rightarrow

\frac{2 x y}{x^{2} + y^{2}} ⩽ - 1 \Rightarrow 2 x y ⩽ - x^{2} - y^{2} \Rightarrow {(x + y)}^{2} ⩽ 0

\Rightarrow x = - y

\frac{2 x y}{x^{2} + y^{2}} ⩾ 1 \Rightarrow 2 x y ⩾ x^{2} + y^{2} \Rightarrow 0 ⩾ {(x - y)}^{2}

\Rightarrow x = y

\Rightarrow\Rightarrow x = \pm y

Commented by Frix last updated on 08/Apr/24

\sec α = \frac{1}{\cos α} \Rightarrow \sec α \in (- \infty, - 1] \cup [1, \infty)

Answered by Frix last updated on 08/Apr/24

\sec θ = \frac{1}{\cos θ}

\cos θ = \frac{x^{2} + y^{2}}{2 x y} = c

x^{2} + y^{2} = 2 c x y

y^{2} - 2 c x y + x^{2} = 0

y = (c \pm \sqrt{c^{2} - 1}) x

y = (\cos θ \pm i \sin θ) x

y \in R \Rightarrow θ = n π \Rightarrow \cos θ = \pm 1 \Rightarrow y = \pm x

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