Question

Eliminate $x,y$ from the equations $x+y=a,x^2+y^2=b^2,x^4+y^4=c^4$.

Answer 1

$x+y=a$ ........ $\left(i\right)$

$x^2+y^2=b^2$
$(x+y{)}^2-2xy=b^2$........ $\left(ii\right)$

$x^4+y^4=c^4$
$\left(\right(x+y{)}^2-2xy{)}^2-2x^2{y}^2=c^4$ ........ $\left(iii\right)$
$(a^2-2xy)=b^2$ ....... [From $\left(i\right)$ and $\left(ii\right)$]
$xy=\frac{a^2-b^2}{2}$
Substitute the value of $x+y,xy$ in the third equation, we get
$\left(\right(a^2-2(a^2-b^2){)}^2-\frac{(a^2-b^2{)}^2}{2})=c^4$
Thus, we eliminated $x,y$ in the above equation
$a^4-2a^2{b}^2-b^4+2c^4=0$ is the required eliminated equation.