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📰 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 $9x^2 - 18x - 16y^2 + 64y = 144$. 📰 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 $oxed{(1, 2)}$. 📰 Why This Hidden Wood Choice Is Revolutionizing Photo Frames In 2024 6073790 📰 5Revealed The Surprisingly Simple Gun Drawing Technique No One Talks About But Everyone Wishes They Knew 3271644 📰 Gikits Hidden Tip You Must Try Before Your Device Crashes 8274351 📰 Stained Glass Secrets Unlocked Found Near You Before Its Too Late 859302 📰 Southampton Ny 8233508 📰 The Epic Build Big Tower In A Dotthis Tiny Square Holds A Gigantic Secret 517942 📰 Best Airport Lounge 2936390 📰 How To Dominate The Frat Scavenger Huntwatch Students Go Wild 6381242 📰 The Fire That Burned Miss Scarletcould It Be Revenge Or Something Far Worse 9149364 📰 Vivaldi 4148541 📰 Discover Why Sixlets Are Taking The Internet By Storm You Wont Believe This 4577877 📰 K Lite Lite 254702 📰 Her Whispered Admiration Over His Perfectly Cut Aquiline Nose 6516670 📰 Grammar Cases 7691459 📰 Insomnia For Mac 4432635