Answer
$2x-y-7=0$
Work Step by Step
The equation of a line in slope-intercept form is:
$$y=mx+b,\tag1$$
where $m$ is the slope and $b$ in the $y$-intercept.
We are given:
- the line is perpendicular to the line passing through P(1,1)\text{ and }Q(5,-1)
- $R(-2,-11)$ belongs to the line.
Because the line is perpendicular to the line passing through $P$ and $Q$, it means the product of their slopes is $-1$. We determine the slope $m_1$ of the line passing through $P$ and $Q$:
$$\begin{align*}
m_1&=\dfrac{y_Q-y_P}{x_Q-x_P}\\
&=\dfrac{-1-1}{5-1}\\
&=-\dfrac{1}{2}.
\end{align*}$$
Determine the slope $m$ of the line we are looking for:
$$\begin{align*}
m_1m&=-1\\
-\dfrac{1}{2}m&=-1\\
m&=2.
\end{align*}$$
We substitute $m=2$ in Eq. $(1)$:
$$y=2x+b.\tag2$$
Determine $b$ using the coordinates of the point $R$ in Eq. $(2)$:
$$\begin{align*}
-11&=2(-2)+b\\
b&=-7.
\end{align*}$$
The equation of the line is:
$$y=2x-7.$$
We write it in the general form $Ax+By+C=0$:
$$2x-y-7=0.$$
