Question

Eliminate $p,q$ from the equations $x(p+q)=y,p-q=k(1+pq),xpq=a$.

Answer 1

$p+q=\frac{y}{x}$ ...... $\left(i\right)$
$p-q=k(1+pq)$ ...... $\left(i\right)$

$xpq=a$ ...... $\left(iii\right)$
Squaring the equations $\left(i\right)$,$\left(ii\right)$ and subtract them,

$(p+q{)}^2-(p-q{)}^2={\left(\frac{y}{x}\right)}^2-\left(k\right(1+pq){)}^2$
$4pq={\left(\frac{y}{x}\right)}^2-\left(k\right(1+pq){)}^2$
Substitute $pq$ from $\left(iii\right)$ in the above equation, we get

$4\frac{a}{x}={\left(\frac{y}{x}\right)}^2-{\left(k(1+\frac{a}{x})\right)}^2$
$4ax=y^2-k^2(x+a{)}^2$
Thus, we eliminated $p,q$ from the given set of equations.