Discussion
The real variables \(x, y, z\), and the real constants \(p, q, r\) satisfy
\(\dfrac{x}{pq - r^{2}} = \dfrac{y}{qr - p^{2}} = \dfrac{z}{rp - q^{2}}\)
Given that the denominators are non-zero, the value of \(px + qy + rz\) is
Answer: A
The value of px + qy + rz is 0.
Call the common value of the three ratios k, so x = k(pq - r2), y = k(qr - p2) and z = k(rp - q2). Then:
\[ \begin{aligned} px + qy + rz &= k[p^{2}q - pr^{2} + q^{2}r \\ &\quad - qp^{2} + r^{2}p - rq^{2}] \\ &= 0 \end{aligned} \]
Every term cancels with another (p2q with -qp2, -pr2 with r2p, q2r with -rq2), whatever k is. So the answer cannot be 1, pqr or p2 + q2 + r2, which would all depend on the constants.
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