Question

Eliminate $m,n$ between the equations $mx-ny=a(m^2-n^2),nx+my=2amn,m^2+n^2=1$.

Answer 1

$mx-ny=a(m^2-n^2)$ ..... $\left(i\right)$

$nx+my=2amn$..... $\left(ii\right)$

$m^2+n^2=1$ ..... $\left(iii\right)$

Squaring the equations $\left(i\right)$, $\left(ii\right)$ and then adding, we get

$(mx-ny{)}^2+(nx+my{)}^2=a^2(m^2-n^2{)}^2+(2amn{)}^2$
$(m^2+n^2)x^2+(m^2+n^2)y^2=a^2(m^2-n^2{)}^2+4a^2{m}^2{n}^2$
$x^2+y^2=a^2(m^2+n^2{)}^2$
$x^2+y^2=a^2\left(1\right)$
Therefore, the eliminated equation is
$x^2+y^2=a^2$