Question

The number of values of t for which the system of equations $(a-t)k+by+c=0$, $bx+(c-t)y+az=0$, $cx+ay+(b-t)z=0$ has non-trivial solution is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A $1$

for the non-trivial solutions

$\left|\begin{array}{ccc}a-t& b& c\\ b& c-t& a\\ c& a& b-t\end{array}\right|=0$

$\left|\begin{array}{ccc}a+b+c-t& a+b+c-t& a+b+c-t\\ b& c-t& a\\ c& a& b-t\end{array}\right|=0$

$t=a+b+cor\left|\begin{array}{ccc}1& 1& 1\\ b& c-t& a\\ c& a& b-t\end{array}\right|=0⟹t^2=(c-a{)}^2-(a-b\left)\right(b-c)=a^2+b^2+c^2-ab-bc-ca$