Let (a,b) be the solution of the system
`[(x,y)][(1,3),(5,1)]=[(2,1)]`
If `alpha` and `beta` are roots of `ax^(2) +2bx -(a+b) =0` then the equation whose are `alpha beta` and `(1)/(alpha)+(1)/(beta)` is :

Options

A. `12x^(2)+17x-40=0`
B. `12x^(2)-53x+56=0`
C. `12x^(2)-53x+56=0`
D. `9x^(2)+54x+80=0`

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and break down the process clearly.

### Step 1: Solve the matrix equation
We need to solve the matrix equation:

\[ \begin{pmatrix} x & y \end{pmatrix} \begin{pmatrix} 1 & 3 \\ 5 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \end{pmatrix} \]

This expands to the system of equations:

1. \( x + 5y = 2 \) (Equation 1)
2. \( 3x + y = 1 \) (Equation 2)