If ${\mathrm{ax}}_1^2+{\mathrm{by}}_1^2+{\mathrm{cz}}_1^2={\mathrm{ax}}_2^2+{\mathrm{by}}_2^2+{\mathrm{cz}}_2^2={\mathrm{ax}}_3^2+{\mathrm{by}}_3^2+{\mathrm{cz}}_3^2=\mathrm{\lambda }$ and ${\mathrm{ax}}_1{\mathrm{x}}_2+{\mathrm{by}}_1{\mathrm{y}}_2+{\mathrm{cz}}_1{\mathrm{z}}_2={\mathrm{ax}}_2{\mathrm{x}}_3+{\mathrm{by}}_2{\mathrm{y}}_3+{\mathrm{cz}}_2{\mathrm{z}}_3={\mathrm{ax}}_3{\mathrm{x}}_1+{\mathrm{by}}_3{\mathrm{y}}_1+{\mathrm{cz}}_3{\mathrm{z}}_1=\mathrm{\mu }$ and $\left|\begin{array}{l}{\mathrm{x}}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{y}}_1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{z}}_1\\ {\mathrm{x}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{y}}_2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{z}}_2\\ {\mathrm{x}}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{y}}_3\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{z}}_3\end{array}\right|={(\mathrm{\lambda }-\mathrm{\mu })}^{\mathrm{\beta }}\left(\frac{\mathrm{\lambda }+\mathrm{\theta \mu }}{\mathrm{abc}}\right)$then the value of $\mathrm{\theta }+\mathrm{\beta }$ is

$\begin{array}{l}\mathrm{LHS}=\left|\begin{array}{lll}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{z}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{z}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{z}}_3\end{array}\right|\times \left|\begin{array}{lll}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{z}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{z}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{z}}_3\end{array}\right|\\ =\frac{1}{\mathrm{abc}}\left|\begin{array}{lll}{\mathrm{ax}}_1& {\mathrm{by}}_1& {\mathrm{cz}}_1\\ {\mathrm{ax}}_2& {\mathrm{by}}_2& {\mathrm{cz}}_2\\ {\mathrm{ax}}_3& {\mathrm{by}}_3& {\mathrm{cz}}_3\end{array}\right|\times \left|\begin{array}{ccc}{\mathrm{x}}_1& {\mathrm{y}}_1& {\mathrm{z}}_1\\ {\mathrm{x}}_2& {\mathrm{y}}_2& {\mathrm{z}}_2\\ {\mathrm{x}}_3& {\mathrm{y}}_3& {\mathrm{z}}_3\end{array}\right|\\ =\left|\begin{array}{ccc}a{x}_1^2+b{y}_1^2+c{z}_1^2& a{x}_1{x}_2+b{y}_1{y}_2+c{z}_1{z}_2& a{x}_3{x}_1+b{y}_3{y}_1\\ a_1{x}_2+b{y}_1{y}_2+c{z}_1{z}_2& a{x}_2^2+b{y}_2^2+c{z}_2^2& a{x}_2{x}_3+b{y}_2{y}_3+c{z}_2{z}_3\\ a_3{x}_1+b{y}_3{y}_1+c{z}_3{z}_1& a{x}_2{x}_3+b{y}_2{y}_3+c{z}_2{z}_3& a{x}_3^2+b{y}_3^2+c{z}_3^2\end{array}\right|\\ =\frac{1}{\mathrm{abc}}\left|\begin{array}{lll}\mathrm{\lambda }& \mathrm{\mu }& \mathrm{\mu }\\ \mathrm{\mu }& \mathrm{\lambda }& \mathrm{\mu }\\ \mathrm{\mu }& \mathrm{\mu }& \mathrm{\lambda }\end{array}\right|\\ C_1\to \hspace{0.17em}{C}_1+C_2+C_3\\ =\frac{1}{\mathrm{abc}}\left|\begin{array}{lll}\mathrm{\lambda }+2\mathrm{\mu }& \mathrm{\mu }& \mathrm{\mu }\\ \mathrm{\lambda }+2\mathrm{\mu }& \mathrm{\lambda }& \mathrm{\mu }\\ \mathrm{\lambda }+2\mathrm{\mu }& \mathrm{\mu }& \mathrm{\lambda }\end{array}\right|\\ =\frac{\left(\mathrm{\lambda }+2\mathrm{\mu }\right)}{\mathrm{abc}}\left|\begin{array}{lll}1& \mathrm{\mu }& \mathrm{\mu }\\ 1& \mathrm{\lambda }& \mathrm{\mu }\\ 1& \mathrm{\mu }& \mathrm{\lambda }\end{array}\right|\\ R_2\to R_2-R_1 , R_3\to R_3-R_1\\ =\frac{\left(\mathrm{\lambda }+2\mathrm{\mu }\right)}{\mathrm{abc}}\left|\begin{array}{lll}1& \mathrm{\mu }& \mathrm{\mu }\\ 0& \mathrm{\lambda }-\mathrm{\mu }& 0\\ 0& 0& \mathrm{\lambda }-\mathrm{\mu }\end{array}\right|\\ Expanding determinant along C_{1 }is\\ =\frac{1}{\mathrm{abc}}(\mathrm{\lambda }-\mathrm{\mu }{)}^2(\mathrm{\lambda }+2\mathrm{\mu })\\ \Rightarrow \mathrm{\theta }=2,\mathrm{\beta }=2\Rightarrow \mathrm{\theta }+\mathrm{\beta }=4\end{array}$