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๐Ÿ“ฐ 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 $oxed{\dfrac{144}{29}}$. ๐Ÿ“ฐ Question: Find the center of the hyperbola $9x^2 - 18x - 16y^2 + 64y = 144$. ๐Ÿ“ฐ Solution: Complete the square for $x$ and $y$. For $x$: $9(x^2 - 2x) = 9[(x - 1)^2 - 1] = 9(x - 1)^2 - 9$. For $y$: $-16(y^2 - 4y) = -16[(y - 2)^2 - 4] = -16(y - 2)^2 + 64$. Substitute back: $9(x - 1)^2 - 9 - 16(y - 2)^2 + 64 = 144$. Simplify: $9(x - 1)^2 - 16(y - 2)^2 = 89$. The center is at $(1, 2)$. Thus, the center is $oxed{(1, 2)}$. ๐Ÿ“ฐ From Billionaire Meetups To Luxy Dating Your Step By Step Guide Now 4123670 ๐Ÿ“ฐ Glamping Economy 9534355 ๐Ÿ“ฐ Detox Water And Lemon 673405 ๐Ÿ“ฐ You Wont Believe These Jaw Dropping Free Fps Shooting Games Higher Rated Now 8241785 ๐Ÿ“ฐ Dare To Try The Ultimate Ds Emulator Watch Classic Games Come To Life Today 2876007 ๐Ÿ“ฐ Unlock Hypnotic Driving Experiences With Top Down Simulator Games Online 5940758 ๐Ÿ“ฐ Calories Apple Honeycrisp 7905871 ๐Ÿ“ฐ Lifesaver In The Block Diamond Finder Minecraft Tool That Finds Diamonds Instantly 1051661 ๐Ÿ“ฐ Nothing Compares 7028635 ๐Ÿ“ฐ Calories Mcnuggets 20 Piece 4419670 ๐Ÿ“ฐ The Gilded Age Season 2 Exposes The Real Betrayals Beneath The Glamour 2681769 ๐Ÿ“ฐ Eat Like A Local In Claremont Top Restaurants Blinking On Foodie Radar 6755865 ๐Ÿ“ฐ Rathalos Strikes Back The Legendary Dragon You Thought Was Myth 3294890 ๐Ÿ“ฐ Chipotle Bogo Today 9003605 ๐Ÿ“ฐ Why Nly Stock Is The Hottest Investment In Townwatch This Trend Explode 4835718