Let the plane $\mathrm{P}:\overrightarrow{\mathrm{r}}\cdot \overrightarrow{\mathrm{a}}=\mathrm{d}$ contain the line of intersection of two planes $\overrightarrow{\mathrm{r}}\cdot (\hat{\mathrm{i}}+3\hat{\mathrm{j}}-\hat{\mathrm{k}})=6\text{ and }\overrightarrow{\mathrm{r}}\cdot (-6\hat{\mathrm{i}}+5\hat{\mathrm{j}}-\hat{\mathrm{k}})=7$. If the plane P passes through the point $\left(2, 3, \frac{1}{2}\right)$, then the value of $\frac{|13\overrightarrow{\mathrm{a}}{|}^2}{{\mathrm{d}}^2}$ is equal to

Equation of the plane passing through the line of intersection of the planes

${P}_1:\overrightarrow{\mathrm{r}}\left(\right(\widehat{\mathrm{i}}+3\widehat{\mathrm{j}}-\widehat{\mathrm{k}})=6$  $\mathrm{and} {\mathrm{P}}_2:\overrightarrow{\mathrm{r}}\cdot (-6\widehat{\mathrm{i}}+5\widehat{\mathrm{j}}-\widehat{\mathrm{k}})=7\text{ is }$${\mathrm{P}}_1+\lambda {\mathrm{P}}_2=0$

$(\overline{\mathrm{r}}\cdot (\widehat{\mathrm{i}}+3\widehat{\mathrm{j}}-\widehat{\mathrm{k}})-6)+\lambda (\overline{\mathrm{r}}\cdot (-6\widehat{\mathrm{i}}+5\widehat{\mathrm{j}}-\widehat{\mathrm{k}})-7)=0$ and it passes through the point 

$\left(2,3,\frac{1}{2}\right)$$\Rightarrow \lambda =1$$\text{ Equation of plane is }\overline{\mathrm{r}}\cdot (-5\widehat{\mathrm{i}}+8\widehat{\mathrm{j}}-2\widehat{\mathrm{k}})=13$

$|\overrightarrow{a}{|}^2=25+64+4=93;d=13$$\therefore \frac{|13\overline{a}{|}^2}{d^2}=93$