System of equations matrices

Question

Please help me in slicing this question.Given a matrix and need to find The locus of all (a,b) € R2 such that this system has atleast TWO DISTINCT solutions for (x1,x2,x3,x4) is ____

Options are Parabola; straight line ; entire plane of R2 ; a point

Answer 1

As commented, your homogeneous system has at least two different solutions iff the coefficients' matrix is singular iff its determinant is zero:

$$\begin{vmatrix}1&2&3&4\\ 5&6&7&8\\ a&9&b&10\\ 6&8&10&13\end{vmatrix}\stackrel{\begin{cases}R_2-5R_1\\R_3-aR_1\\R_4-6R_1\\{}&{}\end{cases}}=\begin{vmatrix}1&2&3&4\\ 0&-4&-8&-12\\ 0&9-2a&b-3a&10-4a\\ 0&-4&-8&-11\end{vmatrix}=$$$${}$$

$$=\begin{vmatrix} -4&-8&-12\\ 9-2a&b-3a&10-4a\\ -4&-8&-11\end{vmatrix}\stackrel{R_3-R_1}=\begin{vmatrix} -4&-8&-12\\ 9-2a&b-3a&10-4a\\ 0&0&1\end{vmatrix}=$$$${}$$

$$=\begin{vmatrix} -4&-8\\ 9-2a&b-3a&\end{vmatrix}=12a-4b+72-16a\stackrel?=0\iff a+b=18$$