Consider the following system of equations: $$ \left[\begin{array}{cccc} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ a & 9 & b & 10 \\ 6 & 8 & 10 & 13 \end{array}\right]\left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] $$ The locus of all $(a, b) \in \mathbb{R}^{2}$ such that this system has at least two dis- tinct solutions for $\left(x_{1}, x_{2}, x_{3}, x_{4}\right)$ is
My Approach:
Here equation will be $$ax1+bx2+cx3+dx4=0$$
Here no polynomial degree of x
and also there are atleast 2 distinct point
and the line passes through these points
$So$, $this$ $should$ $be$ $a$ $straight$ $line$
Hint: the determinant of the matrix is easily calculated: $-4a-4b+72$. Now there are more than one solution iff that determinant equals zero.