Let \(L_1\) be the line of intersection of the planes given by the equation- \(2 x+3 y+z=4 \text { and } x+2 y+z=5 .\)
Let \(L_2\) be the line passing through the point \(P(2,-1,3)\) and parallel to \(L_1\). Let \(M\) denote the plane given by the equation- \(2 x+y-2 z=6\)
Suppose that the line \(L_2\) meets the plane \(M\) at the point \(Q\). Let \(R\) be the foot of the perpendicular drawn from \(P\) to the plane \(M\).
Then which of the following statements is (are) TRUE?

Let \(L_1: \vec{r}=\vec{a}+t \vec{b}\)
\(\overrightarrow{\mathrm{b}}=\left|\begin{array}{ccc}\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\ 2 & 3 & 1 \\ 1 & 2 & 1\end{array}\right|=\hat{\mathrm{i}}(1)-\hat{\mathrm{j}}(1)+\hat{\mathrm{k}}(1)\)
Dr's of \(L_1: <1,-1,1>\)
\(\begin{aligned}
& L_1: \frac{x+1}{1}=\frac{y-0}{-1}=\frac{z-6}{1} \\
& \& L_2: \frac{x-2}{1}=\frac{y+1}{-1}=\frac{z-3}{1}=\lambda \\
& M: 2 x+y-2 z-6=0
\end{aligned}\)
Let point on \(\mathrm{L}_2(\lambda+2,-\lambda-1, \lambda+3)\)
\(\begin{aligned}
& 2(\lambda+2)-\lambda-1-2 \lambda-6-6=0 \\
& 2 \lambda+4-3 \lambda-13=0 \\
& \lambda=-9 \\
& \therefore Q(-7,8,-6)
\end{aligned}\)

Line PR: \(\frac{x-2}{2}=\frac{y+1}{1}=\frac{z-3}{-2}=\mu\)
\(\mathrm{R}(2 \mu+2, \mu-1,-2 \mu+3)\)
Put in plane
\(\begin{aligned}
& 2(2 \mu+2)+\mu-1-2(-2 \mu+3)-6=0 \\
& 4 \mu+4+\mu-1+4 \mu-6-6=0 \\
& 9 \mu-9=0 \Rightarrow \mu=1 \\
& \mathrm{R}(4,0,1) \\
& \mathrm{P}(2,-1,3) \& \mathrm{Q}(-7,8,-6) \\
& \mathrm{PQ}=\sqrt{81+81+81}=9 \sqrt{3} \\
& \mathrm{Q}(-7,8,-6) \& \mathrm{R}(4,0,1)
\end{aligned}\)
\(\mathrm{QR}=\sqrt{121+64+49}=\sqrt{234}\)
Area ( \(\triangle \mathrm{PQR}\) )
\(\begin{aligned}
& =\frac{1}{2}|\overrightarrow{\mathrm{QP}} \times \overrightarrow{\mathrm{QR}}| \\
& =\frac{1}{2}\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
9 & -9 & 9 \\
11 & -8 & 7
\end{array}\right| \\
& =\frac{3}{2} \sqrt{234} \\
& \overrightarrow{\mathrm{PQ}}=-9 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}-9 \hat{\mathrm{k}}=-9(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}) \\
& \overrightarrow{\mathrm{PR}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}
\end{aligned}\)
\(\begin{aligned} & \cos \theta=\left|\frac{\overrightarrow{\mathrm{PQ}} \cdot \overrightarrow{\mathrm{PR}}}{\mathrm{PQ} \cdot \mathrm{PR}}\right| \\ & =\frac{9}{9 \sqrt{3} \times 3}=\frac{1}{3 \sqrt{3}} \\ & \theta=\cos ^{-1}\left(\frac{1}{3 \sqrt{3}}\right)\end{aligned}\)