If the system of equations

$2 x+y-z=5$

$2 x-5 y+\lambda z=\mu$

$x+2 y-5 z=7$

has infinitely many solutions, then $(\lambda+\mu)^{2}+(\lambda-\mu)^{2}$ is equal to

$$ \begin{aligned} & 2 x+y-z=5 \\ & 2 x-5 y+\lambda z=\mu \\ & x+2 y-5 z=7 \end{aligned} $$ <br/><br/>For infinite solution $\Delta=0=\Delta_1=\Delta_2=\Delta_3$ <br/><br/>$$ \Delta=\left|\begin{array}{ccc} 2 & 1 & -1 \\ 2 & -5 & \lambda \\ 1 & 2 & -5 \end{array}\right|=0 $$ <br/><br/>$$ \begin{aligned} \Rightarrow& 2(25-2 \lambda)-(-10-\lambda)-(4+5)=0 \\\\ \Rightarrow& 50-4 \lambda+10+\lambda-9=0 \\\\ \Rightarrow& 51=3 \lambda \Rightarrow \lambda=17 \end{aligned} $$ <br/><br/>$$ \Delta_3=\left|\begin{array}{ccc} 5 & 2 & 1 \\ \mu & 2 & -5 \\ 7 & 1 & 2 \end{array}\right|=0 $$ <br/><br/>$$ \begin{aligned} \Rightarrow & 2(-35-2 \mu)-(14-\mu)+5(4+5)=0 \\\\ \Rightarrow & -70-4 \mu-14+\mu+45=0 \\\\ \Rightarrow & -3 \mu=39 \\\\ \Rightarrow & -\mu=13 \end{aligned} $$ <br/><br/>Now $(\lambda+\mu)^2+(\lambda-\mu)^2$ <br/><br/>$$ \begin{aligned} & (17+13)^2+(17-13)^2 \\\\ & 900+16 \\\\ & =916 \end{aligned} $$