GATE Mechanical 2024 | GA | Question: 4

Question

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

\[
\frac{x}{p q-r^{2}}=\frac{y}{q r-p^{2}}=\frac{z}{r p-q^{2}}
\]

Given that the denominators are non-zero, the value of $p x+q y+r z$ is

• A. $0$
• B. $1$
• C. $p q r$
• D. $p^{2}+q^{2}+r^{2}$

Answer 1

Let, $K$ be a real number such that

\[
\frac{x}{p q-r^{2}}=\frac{y}{q r-p^{2}}=\frac{z}{r p-q^{2}}=K
\]

$\implies$ \[
x=K*(pq-r^{2})
\]

\[
y=K*(qr-p^{2})
\]

\[
z=K*(rp-q^{2})
\]

$\therefore px + qy + rz = K*[\color{red}{p^2q}\color{green}{-r^2p}\color{blue}{+q^2r}\color{red}{-p^2q}\color{green}{+r^2p}\color{blue}{-q^2r}\color{black}{] = K*0=0}$

Hence, Option A is the correct option.

Answer 2

Step by step explanations 

Option A is correct