Nature of Roots

How many ordered pairs of integers $(x,y)$ satisfy the equation $x^2+y^2=2(x+y)+xy$?

If $a>b>0$ and $a^3+b^3+27ab=729$, then the quadratic equation $a{x}^2+bx-9=0$ has roots $\alpha \& \beta$, $(\alpha <\beta )$. Find the value of $4\beta -a\alpha$.

Find the set of all real values of $\mathrm{\lambda }$ such that the root of the equation $x^2+2\left(a+b+c\right)x+3\lambda \left(ab+bc+ca\right)=0$ are always real for any choice of $a, b, c$ (Where $a, b, c$ represents sides of scalene triangle).

If the equation $\left|x^2-5x+6\right|-\lambda x+7\lambda =0$ has exactly three solutions, then $\lambda$ is equal to

Let $a, b, c$ and $m\in R^{+},$ the possible value of $m$ (independent of $a, b$ and $c$) for which at least one of the following equations have real roots, is

$a{x}^2+bx+cm=0$, $b{x}^2+cx+am=0$, $c{x}^2+ax+bm=0$.