If the system of equations [ aligned & 2 x+ y+3 z=5 & 3 x+2 y-z=7 & 4 x+5 y+ z=9 aligned ]…

Question

If the system of equations

$$ \begin{aligned} & 2 x+\lambda y+3 z=5 \ & 3 x+2 y-z=7 \ & 4 x+5 y+\mu z=9 \end{aligned} $$

has infinitely many solutions, then $\left(\lambda^2+\mu^2\right)$ is equal to :

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
& 2 x+\lambda y+3 z=5 \
& 3 x+2 y-z=7 \
& 4 x+5 y+\mu z=9
\end{aligned}\
\&\text { For infinite solutions } \Rightarrow \Delta=0=\Delta_1=\Delta_2=\Delta_3
\end{aligned}$$

$$\begin{aligned}
& \Delta=\left|\[\begin{array}{lll} 2 & \lambda & 3 \ 3 & 2 & -1 \ 4 & 5 & \mu \end{array}\end{aligned}$$\right|=0 \
\& \Rightarrow-4 \lambda-3 \lambda \mu+4 \mu+31=0 \
\& \Delta\_1=\left|$$\begin{array}{ccc} 5 & \lambda & 3 \ 7 & 2 & -1 \ 9 & 5 & \mu \end{array}$$\right|=0 \Rightarrow-9 \lambda-7 \lambda \mu+10 \mu+76=0 \
\& \Delta\_2=\left|$$\begin{array}{ccc} 2 & 3 & 5 \ 3 & -1 & 7 \ 4 & \mu & 9 \end{array}$$\right|=0 \Rightarrow \mu+5=0 \Rightarrow \mu=-5 \
\& \Delta\_3=\left|$$\begin{array}{lll} 2 & \lambda & 5 \ 3 & 2 & 7 \ 4 & 5 & 9 \end{array}$$\right|=0 \Rightarrow \lambda+1=0 \Rightarrow \lambda=-1
\]

$$\begin{aligned}
&\therefore \text { For infinite solution } \mu=-5 \text { and } \lambda=-1\
&\begin{aligned}
\text { Now } \mu^2+\lambda^2 & =25+1 \
& =26
\end{aligned}
\end{aligned}$$

Solutions · 1