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GATE 2024 PH – Question 4

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

The real variables $x,y,z$ and constants $p,q,r$ satisfy $x/(pq-r^2)=y/(qr-p^2)=z/(rp-q^2)$. Given nonzero denominators, $px+qy+rz$ 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 $k$:
$$\frac{x}{pq-r^2}=\frac{y}{qr-p^2}=\frac{z}{rp-q^2}=k .$$

So $x=k(pq-r^2)$, $y=k(qr-p^2)$ and $z=k(rp-q^2)$.

**Compute $px+qy+rz$:**
$$px+qy+rz=k\left[p(pq-r^2)+q(qr-p^2)+r(rp-q^2)\right]$$
$$=k\left[p^2q-pr^2+q^2r-p^2q+r^2p-q^2r\right]=k\times0=\mathbf{0}.$$

Every term cancels in pairs ($p^2q$ with $-p^2q$, $-pr^2$ with $r^2p$, and $q^2r$ with $-q^2r$). Answer **0** (A).