Question

If the equations $ax+hy+gz=0$, $hx+by+fz=0,gx+fy+cz=\lambda z$ are solvable then the value $\lambda$ is

• A. $0$
• B. $\frac{abc+2gfh-a{f}^2-b{g}^2-c{h}^2}{ab-h^2}$
• C. $a{b}_C+2fgh-a{f}^2-b{g}^2-c{h}^2$
• D. $\frac{a{f}^2+b{g}^2+c{h}^2-abc-2fgh}{ab-h^2}$

Answer 1

Answer: (B)

$\frac{abc+2gfh-a{f}^2-b{g}^2-c{h}^2}{ab-h^2}$

Given $an+hy+gz=0$
$hx+by+fz=0$
$gx+fy+cz=\lambda z$
has solution so,
$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c-\lambda \end{array}\right|=0$
By solving determinant, we get
$\lambda =\frac{abc+2fgh-a{f}^2-b{g}^2-c{h}^2}{ab-h^2}$