2019-II-37

From $\text{①}$, we have

$\begin{array}{rcl}3x–y–2& =& 0\\ y& =& 3x–2\end{array}$

Sub. $y=3x–2$ into $\text{②}$, we have

$\begin{array}{rcl}5x^2+5(3x–2{)}^2+kx+4(3x–2)–20& =& 0\\ 5x^2+45x^2-60x+20+kx+12x–8–20& =& 0\\ 50x^2+(k–48)x–8& =& 0\end{array}$

Since the $x$-coordinate of the mid-point of $PQ$ is $2$, we have

$\begin{array}{rcl}{\displaystyle \frac{\text{sum of the roots}}{2}}& =& 2\\ {\displaystyle \frac{-(k-48)}{50}}\times {\displaystyle \frac{1}{2}}& =& 2\\ -(k–48)& =& 200\\ k–48& =& -200\\ k& =& -152\end{array}$