图中显示了三个点 $P(-6, 0)$、$Q(4k, k-2)$ 和 $R(8, 5)$，其中 $PQ$ 垂直于 $QR$ 且 $k < 0.$

1. Find the gradients of the line segments $PQ$ and $QR$ in terms of $k$.

Gradient of $PQ$, $m_{PQ} = \frac{(k-2) - 0}{4k - (-6)} = \frac{k-2}{4k+6}$

Gradient of $QR$, $m_{QR} = \frac{5 - (k-2)}{8 - 4k} = \frac{7-k}{8-4k}$

2. Apply the perpendicularity condition to form an equation in $k$.

Since $PQ \perp QR$, $m_{PQ} \times m_{QR} = -1$.

$\left(\frac{k-2}{4k+6}\right) \left(\frac{7-k}{8-4k}\right) = -1$

$(k-2)(7-k) = -(4k+6)(8-4k)$

$7k - k^2 - 14 + 2k = -(32k - 16k^2 + 48 - 24k)$

$-k^2 + 9k - 14 = 16k^2 - 8k - 48$

3. Solve the resulting quadratic equation for $k$ and reject the invalid root.

$17k^2 - 17k - 34 = 0$

$k^2 - k - 2 = 0$

$(k-2)(k+1) = 0$

$k = 2$ or $k = -1$

Since it is given that $k < 0$, we reject $k = 2$.

Therefore, $k = -1$ (shown).

1. Find the coordinates of the midpoint of $PR$.

Midpoint of $PR = \left( \frac{-6 + 8}{2}, \frac{0 + 5}{2} \right) = (1, 2.5)$

2. Find the gradient of the line segment $PR$.

Gradient of $PR = \frac{5 - 0}{8 - (-6)} = \frac{5}{14}$

3. Determine the gradient of the perpendicular bisector.

Gradient of perpendicular bisector $= -\frac{1}{\frac{5}{14}} = -\frac{14}{5} = -2.8$

4. Find the equation of the perpendicular bisector and determine its $y$-intercept.

Equation of the perpendicular bisector:
$y - 2.5 = -2.8(x - 1)$
Since $S$ lies on the $y$-axis, $x = 0$:
$y - 2.5 = -2.8(0 - 1)$
$y = 2.5 + 2.8 = 5.3$
Coordinates of $S$ are $(0, 5.3)$.

1. State the coordinates of the vertices and set up the shoelace formula

From earlier parts, the coordinates are $P(-6, 0)$, $Q(-4, -3)$, $R(8, 5)$, and $S(0, 5.3)$.
Arranging the vertices in counter-clockwise order, the area of quadrilateral $PQRS$ is given by:
$\text{Area} = \frac{1}{2} \left| \begin{matrix} -6 & -4 & 8 & 0 & -6 \\ 0 & -3 & 5 & 5.3 & 0 \end{matrix} \right|$

2. Evaluate the shoelace formula to find the area

$\text{Area} = \frac{1}{2} | [(-6)(-3) + (-4)(5) + (8)(5.3) + (0)(0)] - [(0)(-4) + (-3)(8) + (5)(0) + (5.3)(-6)] |$
$= \frac{1}{2} | (18 - 20 + 42.4 + 0) - (0 - 24 + 0 - 31.8) |$
$= \frac{1}{2} | 40.4 - (-55.8) |$
$= \frac{1}{2} (96.2) = 48.1$