If x : y : z = 2 : 3 : 4, what is the value of \(\left( \frac{{{x}^{2}}+2{{y}^{2}}+{{z}^{2}}}{3{{x}^{2}}-2{{y}^{2}}+4{{z}^{2}}} \right)?\)
19/29
The correct answer is 19/29.
In this problem, all terms in the numerator (x^2, y^2, z^2) and denominator (x^2, y^2, z^2) are of degree 2. When we substitute xk, yk, zk, each term becomes of degree 2 in k (e.g., (xk)^2 = x^2k^2). This uniformity in degree ensures that the k^2 term can be factored out from both the numerator and denominator and cancelled, giving a numerical result independent of the specific value of k.
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