Question

$x,y,z$ not all zeros and the equations $x+y+z=0$, $(1+a)x+(2+a)$$y-8z$ $=0,x-(1+a)y+(2+a)z=0$ have non-trivial solution then $a=$

• A. $2+\sqrt{15}$
• B. $3-\sqrt{15}$
• C. $\sqrt{15}$
• D. $-4$

Answer 1

The correct option is A $2+\sqrt{15}$

$\Delta =\left|\begin{array}{ccc}1& 1& 1\\ 1+a& 2+a& -8\\ 1& -(1+a)& (2+a)\end{array}\right|$
$C_2\to C_2-C_1$
$\Delta =\left|\begin{array}{ccc}1& 0& 1\\ 1+a& 1& -8\\ 1& -(2+a)& (2+a)\end{array}\right|$
$\Delta =(2+a)-8(2+a)-(2+a)(1+a)-1$
$=-14-7a-2-a^2-3a-1a$
$=-a^2-10a-17$
$\Delta =0$
$\therefore a^2+10+17=0$
$\therefore a=2\pm \sqrt{15}$