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Resonant Sounds | PDF

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Resonant Sounds | PDF
Another word for RESONANT > Synonyms & Antonyms
Resonant - Synonym, Antonym, and Examples - Grammar Diary
Resonant Synonyms and Resonant Antonyms. Similar and opposite words for ...
Resonant Synonyms and Resonant Antonyms. Similar and opposite words for ...

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Question: Let $x, y, z$ be positive real numbers such that $2x + 3y + 4z = 12$. Find the minimum value of $x^2 + y^2 + z^2$.
Solution: Use the Cauchy-Schwarz inequality: $(2^2 + 3^2 + 4^2)(x^2 + y^2 + z^2) \geq (2x + 3y + 4z)^2$. This gives $29(x^2 + y^2 + z^2) \geq 144$, so $x^2 + y^2 + z^2 \geq rac{144}{29}$. Equality holds when $ rac{x}{2} = rac{y}{3} = rac{z}{4} = k$, leading to $x = 2k$, $y = 3k$, $z = 4k$. Substituting into $2x + 3y + 4z = 12$ gives $4k + 9k + 16k = 29k = 12$, so $k = rac{12}{29}$. Thus, the minimum value is $oxed{\dfrac{144}{29}}$.
Question: Find the center of the hyperbola $9x^2 - 18x - 16y^2 + 64y = 144$.

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