Let the plane be $\vec{r} \cdot \hat{n}=d$
$\therefore$ Perpendicular distance from $\vec{a}$ to this plane $=|\mathrm{d}-\vec{a} \cdot \hat{n}|$ the plane is
$$
\overrightarrow{\mathrm{r}} \cdot(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}})+13-0
$$
or $\overrightarrow{\mathrm{r}} \cdot\left(\frac{3}{13} \hat{\mathrm{i}}+\frac{4}{13} \hat{\mathrm{j}}-\frac{12}{13} \hat{\mathrm{k}}\right)=-\frac{13}{13}$
(i) $\mathrm{d}=\frac{-13}{13}=-1, \quad \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{P} \hat{\mathrm{k}}$
$$
\begin{aligned}
&\overrightarrow{\mathrm{n}}=\frac{3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}}}{\sqrt{3^2+4^2+12^2}}=\frac{3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}}}{\sqrt{169}} \\
&=\frac{3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}}}{13}
\end{aligned}
$$
$\therefore \quad$ Perpendicular distance $|\overrightarrow{\mathrm{d}}-\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{n}}|$
$$
=\left|\frac{-13}{13}-(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{P} \hat{\mathrm{k}}) \cdot\left(\frac{3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}}}{13}\right)\right|=\left|\frac{20-12 \mathrm{P}}{13}\right|
$$
The second point is $(-3,0,1)$ or $(-3 \hat{\mathrm{i}}+\hat{\mathrm{k}})$
$\therefore$ Perpendicular distance of the plane from the point $\therefore 3 \hat{i}+\hat{k}$ is
$$
\begin{aligned}
&=|\mathrm{d}-\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{n}}|=\left|-1-(-3 \hat{\mathrm{i}}+\hat{\mathrm{k}}) \cdot \frac{3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-12 \hat{\mathrm{k}} \mid}{13}\right| \\
&=\left[-1 \frac{-9-12}{13}\right]=\left[-1+\frac{21}{13}\right]=\frac{8}{13}
\end{aligned}
$$
from (i) and (ii)
$$
\left|\frac{20-2 \mathrm{P}}{13}\right|=\frac{8}{13} \text { or } \frac{20-12 \mathrm{P}}{13}=\pm \frac{8}{13}
$$
Taking $+\mathrm{ve}, 20-12 \mathrm{P}=8$
$$
\Rightarrow 12 \mathrm{P}=20-8=12 \quad \therefore \mathrm{P}=1
$$
Taking-ve, $20-12 \mathrm{P}=-8$
$$
\Rightarrow \mathrm{P}=\frac{28}{12}=\frac{7}{3} \Rightarrow \mathrm{P}=1, \frac{7}{3}
$$