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GATE 2024 ME – Question 4

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

The real variables $x$, $y$, $z$, and the real constants $p$, $q$, $r$ satisfy

$$\frac{x}{pq - r^2} = \frac{y}{qr - p^2} = \frac{z}{rp - q^2}$$

Given that the denominators are non-zero, the value of $px + qy + rz$ is

  1. 0
  2. 1
  3. $pqr$
  4. $p^2 + q^2 + r^2$

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Show answer and explanation

Correct answer: (A) 0

Explanation

Let the common ratio be $k$, so $x = k(pq - r^2)$, $y = k(qr - p^2)$ and $z = k(rp - q^2)$. Then $px + qy + rz = k\left[p^2q - pr^2 + q^2r - qp^2 + r^2p - rq^2\right]$. The terms cancel in pairs ($p^2q - qp^2 = 0$, $-pr^2 + r^2p = 0$, $q^2r - rq^2 = 0$), so the sum is 0.