Question

The sum of square of values of $c$ for which the equations
$2x+3y=3$
$(c+2)x+(c+4)y=(c+6)$
$(c+2{)}^{2}x+(c+4{)}^{2}y=(c+6{)}^2$ are consistent, is

Answer 1

Given system of equations are consistent
$\therefore \Delta =0$
$\Delta =\left|\begin{array}{ccc}2& 3& 3\\ c+2& c+4& c+6\\ (c+2{)}^2& (c+4{)}^2& (c+6{)}^2\end{array}\right|=0$

$C_1\to C_1-C_3;C_2\to C_2-C_3\left|\begin{array}{ccc}-1& 0& 3\\ -4& -2& c+6\\ -4(2c+8)& -2(2c+10)& (c+6{)}^2\end{array}\right|=0\Rightarrow \left|\begin{array}{ccc}1& 0& 3\\ 4& 1& c+6\\ 8c+32& 2c+10& (c+6{)}^2\end{array}\right|=0C_1\to C_1-4C_2\Rightarrow \left|\begin{array}{ccc}1& 0& 3\\ 0& 1& c+6\\ -8& 2c+10& (c+6{)}^2\end{array}\right|=0C_3\to C_3-3C_1\Rightarrow \left|\begin{array}{ccc}1& 0& 0\\ 0& 1& c+6\\ -8& 2c+10& (c+6{)}^2+24\end{array}\right|=0\Rightarrow -c^2-10c=0\Rightarrow c=0,c=-10$
Sum of squares of values of $c$ is $0+100=100$