In the figure above, line $\ell$ is tangent to the graph of $y = \frac { 1 } { x ^ { 2 } }$ at point $P$, with coordinates $\left( w , \frac { 1 } { w ^ { 2 } } \right)$, where $w > 0$. Point $Q$ has coordinates $( w , 0 )$. Line $\ell$ crosses the $x$-axis at point $R$, with coordinates $( k , 0 )$.
(a) Find the value of $k$ when $w = 3$.
(b) For all $w > 0$, find $k$ in terms of $w$.
(c) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of $k$ with respect to time?
(d) Suppose that $w$ is increasing at the constant rate of 7 units per second. When $w = 5$, what is the rate of change of the area of $\triangle PQR$ with respect to time? Determine whether the area is increasing or decreasing at this instant.