Discussion
The real variables x, y, z, and the real constants p, q, r satisfy
\[ \begin{aligned} \frac{x}{pq - r^{2}} &= \frac{y}{qr - p^{2}} \\ &= \frac{z}{rp - q^{2}} \end{aligned} \]
Given that the denominators are non-zero, the value of px + qy + rz is
Answer: A
The value of px + qy + rz is 0.
Let each ratio equal k, so \(x = k(pq - r^{2})\), \(y = k(qr - p^{2})\), \(z = k(rp - q^{2})\). Then
\[ \begin{aligned} px + qy + rz &= k[p^{2}q - pr^{2} + q^{2}r - qp^{2} + r^{2}p - rq^{2}] \\ &= k \times 0 = 0 \end{aligned} \]
The six terms cancel in pairs: \(p^{2}q\) with \(-qp^{2}\), \(-pr^{2}\) with \(r^{2}p\), and \(q^{2}r\) with \(-rq^{2}\). The result does not depend on k, so 1, pqr and \(p^{2} + q^{2} + r^{2}\) cannot be right for all values.
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