Let ( α , β ) be the centroid of the triangle formed by the lines 15 x - y = 82 ,   6 x - 5 y…

The given set of equations are:

$15x-y=82$

$6x-5y=-4$

$9x+4y=17$

Now on solving the above equation by taking two at a time and plotting the diagram we get,

 

$(\alpha ,\beta )\equiv \left(\frac{1+5+6}{3},\frac{2-7+8}{3}\right)\equiv (4,1)$

Hence, $\alpha +2\beta =6$ and $2\alpha -\beta =7$

The required equation is $\left(x-6\right)\left(x-7\right)=0$

Equation $x^2-13x+42=0$

Hence this is the required option.