Eliminate $p, q, r$ from the equations $p + q + r = 0, a (q r + r p + p q) = 2 a - x, a p q r = y, q r = 1$ .

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Solution

Given
$p + q + r = 0$ ...... $(i)$
$a (q r + r p + p q) = 2 a - x$ ...... $(i i)$
$a p q r = y$ ...... $(i i i)$
$q r = - 1$ ...... $(i v)$
$\Rightarrow a p = - y$ .... [From $(i i i)$ ]
$q + r = - p$ .... [From $(i)$ ]
$a (- 1 + p (- p)) = 2 a - x$ .... [From $(i i)$ ]
where $p = \frac{- y}{a}$
Therefore, $- a (1 + {(\frac{y}{a})}^{2}) = 2 a - x$
$y^{2} = a (x - 3 a)$ is the required eliminated equation.

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Eliminate p,q,r from the equations p+q+r=0,a(qr+rp+pq)=2a−x,apqr=y,qr=1.

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Eliminate $p, q, r$ from the equations $p + q + r = 0, a (q r + r p + p q) = 2 a - x, a p q r = y, q r = 1$ .