given-that-there-are-real-constant-a-b-c-d-such-the-identity-x-2-2xy-y-2-ax-by-2-cx-dy-2-holds-for-all-x-y-R-this-implies-a-5-b-1-c-0-lt-lt-1-

Question Number 205101 by universe last updated on 08/Mar/24

given that there are real constant a, b, c, d

such the identity

λ x^{2} + 2 xy + y^{2} = {(ax + by)}^{2} + {(cx + dy)}^{2} holds

for all x, y \in R this implies

(a) λ = - 5 (b) λ ⩾ 1 (c) 0 < λ < 1

(d) there is no such λ \in R

Answered by A5T last updated on 08/Mar/24

λ x^{2} + 2 x y + y^{2}

= (a^{2} + c^{2}) x^{2} + (2 a b + 2 c d) x y + (b^{2} + d^{2}) y^{2}

\Rightarrow b^{2} + d^{2} = 1; a b + c d = 1 ⩽ \sqrt{a^{2} + c^{2}} \sqrt{b^{2} + d^{2}} = \sqrt{a^{2} + c^{2}}

\Rightarrow λ = a^{2} + c^{2} ⩾ 1 \Rightarrow (b)

Commented by universe last updated on 08/Mar/24

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