Consider the following system of equations:

$$\begin{bmatrix}1 & 2 & 3 & 4 \\5 & 6 & 7 & 8 \\a & 9 & b & 10 \\6 & 8 & 10 & 13\end{bmatrix}\begin{bmatrix}x_1 \\x_2 \\x_3 \\x_4 \end{bmatrix} = \begin{bmatrix}0 \\0 \\0 \\0 \end{bmatrix}$$
The locus of all $$(a, b) \epsilon R^2$$ such that this system has at least two distinct solutions for $$(x_1, x_2, x_3, x_4)$$ is

Let the above equation be written as  Ax=b

where A = $$\begin{bmatrix}1 & 2 & 3 & 4 \\5 & 6 & 7 & 8 \\a & 9 & b & 10 \\6 & 8 & 10 & 13 \end{bmatrix}$$

x = $$\begin{bmatrix}x_1 \\x_2 \\x_3 \\x_4 \end{bmatrix}$$

b =  $$\begin{bmatrix}0 \\0 \\0 \\0 \end{bmatrix}$$

It is evident that $$x_1=x_2=x_3=x_4=\ 0\ $$ is a solution for the question. 

It is stated that the equation has at least 2 distinct solution which implies infinite solution exists. and thus det(A) = 0.

Calculating det(A) we get det(A) = 4(a+b-18)

Equating it with 0, we get a+b-18=0 which is a straight line equation