If the system of equations 2 x+y-z=5 [2 x-5 y+ z=] x+2 y-5 z=7 has infinitely many solutio…

Question

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

Accepted Answer

$$
\begin{aligned}
& 2 x+y-z=5 \
& 2 x-5 y+\lambda z=\mu \
& x+2 y-5 z=7
\end{aligned}
$$

For infinite solution $\Delta=0=\Delta_1=\Delta_2=\Delta_3$

$$
\Delta=\left|\[\begin{array}{ccc} 2 & 1 & -1 \ 2 & -5 & \lambda \ 1 & 2 & -5 \end{array}$$\right|=0
\]

$$
\begin{aligned}
\Rightarrow& 2(25-2 \lambda)-(-10-\lambda)-(4+5)=0 \\
\Rightarrow& 50-4 \lambda+10+\lambda-9=0 \\
\Rightarrow& 51=3 \lambda \Rightarrow \lambda=17
\end{aligned}
$$

$$
\Delta_3=\left|\[\begin{array}{ccc} 5 & 2 & 1 \ \mu & 2 & -5 \ 7 & 1 & 2 \end{array}$$\right|=0
\]

$$
\begin{aligned}
\Rightarrow & 2(-35-2 \mu)-(14-\mu)+5(4+5)=0 \\
\Rightarrow & -70-4 \mu-14+\mu+45=0 \\
\Rightarrow & -3 \mu=39 \\
\Rightarrow & -\mu=13
\end{aligned}
$$

Now $(\lambda+\mu)^2+(\lambda-\mu)^2$

$$
\begin{aligned}
& (17+13)^2+(17-13)^2 \\
& 900+16 \\
& =916
\end{aligned}
$$

Solutions · 1