1. **State the problem:** We are given a point $P = (1,4,c)$ in 3D space and a line $r$. We want to find the length of the perpendicular from $P$ to the line $r$ and show that the squared length satisfies a certain relation. 2. **Identify the line $r$:** The problem statement is incomplete about the exact parametric form of line $r$. Typically, a line in 3D is given by $\mathbf{r}(t) = \mathbf{a} + t\mathbf{b}$ where $\mathbf{a}$ is a point on the line and $\mathbf{b}$ is the direction vector. 3. **Length of perpendicular from point to line:** The length $d$ of the perpendicular from point $P$ to line $r$ is given by $$ d = \frac{|(\mathbf{P} - \mathbf{a}) \times \mathbf{b}|}{|\mathbf{b}|} $$ where $\times$ is the cross product. 4. **Assuming $r$ passes through $p = (x,y)$ in 2D and extending to 3D:** Since the problem mentions $p = (x,y)$ and $P = (1,4,c)$, we interpret $r$ as a line in 3D passing through $p$ with some direction vector $\mathbf{m}$. 5. **Given the expression to show:** The problem states to show that $$ 1 - \frac{1}{m^2} = 1 $$ which simplifies to a tautology, so likely the problem wants to show the squared length of the perpendicular equals 1. 6. **Conclusion:** Without explicit line $r$ data, the general formula for the length of the perpendicular from $P$ to $r$ is as in step 3. If the problem provides $m$ and $p$, substitute and simplify to verify the length squared equals 1. **Final answer:** The length of the perpendicular from $P$ to $r$ is $$ d = \frac{|(\mathbf{P} - \mathbf{a}) \times \mathbf{b}|}{|\mathbf{b}|} $$ which can be shown to satisfy the given relation when the line parameters are known.