Let $\ell_{1}$ and $\ell_{2}$ be the lines $\vec{r}_{1} = \lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_{2} = (\hat{j}-\hat{k}) + \mu(\hat{i}+\hat{k})$, respectively. Let X be the set of all the planes H that contain the line $\ell_{1}$. For a plane H, let $d(H)$ denote the smallest possible distance between the points of $\ell_{2}$ and H. Let $H_{0}$ be a plane in X for which $d(H_{0})$ is the maximum value of $d(H)$ as H varies over all planes in X. Match each entry in List-I to the correct entries in List-II. | List-I | List-II | | --- | --- | | (P) The value of $d(H_{0})$ is | (1) $\sqrt{3}$ | | (Q) The distance of the point $(0, 1, 2)$ from $H_{0}$ is | (2) $\frac{1}{\sqrt{3}}$ | | (R) The distance of origin from $H_{0}$ is | (3) $0$ | | (S) The distance of origin from the point of intersection of planes $y = z$, $x = 1$ and $H_{0}$ is | (4) $\sqrt{2}$ | | | (5) $\frac{1}{\sqrt{2}}$ | The correct option is:

Let ell₁ and ell₂ be the lines r⃗₁ = λ(î+ĵ+k̂) and r⃗₂ = (ĵ-k̂) + μ(î+k̂), respectively. Let X be the set of all the planes H that contain the line ell₁. For a plane H, let d(H) denote the smallest possible distance between the points of ell₂ and H. Let H₀ be a plane in X for which d(H₀) is the maximum value of d(H) as H varies over all planes in X. Match each entry in List-I to the correct entries in List-II. | List-I | List-II | | -- $\ell_1:\ \vec{r}=\lambda(\hat{i}+\hat{j}+\hat{k}) \Rightarrow \text{through } (0,0,0), \ \vec{d}_1=(1,1,1)$ $\ell_2:\ \vec{r}=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k}) \Rightarrow \text{through } (0,1,-1), \ \vec{d}_2=(1,0,1)$ $X=\{H:\ell_1\subset H\}, \quad d(H)=\min \text{dist}(\ell_2, H)$ $H_0\in X \text{ maximises } d(H)$ $\text{Match (P),(Q),(R),(S) with (1)--(5)}$ $1.\ H:\ \vec{n}\cdot\vec{r}=0,\quad \vec{n}\perp\vec{d}_1$ $2.\ d(H)=0 \text{ unless } \ell_2\parallel H$ $3.\ \text{maximise } d(H)\Rightarrow H_0, \text{ then (P),(Q),(R),(S)}$ $H:\ \vec{n}\cdot\vec{r}=0 \ \text{(through } O)$ $\vec{n}\perp\vec{d}_1 \Rightarrow n_1+n_2+n_3=0$ $\vec{d}_2\cdot\vec{n}=n_1+n_3$ $\vec{d}_2\cdot\vec{n}\ne 0 \Rightarrow \ell_2 \text{ meets } H \Rightarrow d(H)=0$ $d(H)\ne 0 \Rightarrow n_1+n_3=0$ $n_2=0,\quad \vec{n}\parallel(1,0,-1)$ $H_0:\ x-z=0$ $d(H_0)=\dfrac{|\vec{n}\cdot\overrightarrow{OA}|}{|\vec{n}|},\quad A(0,1,-1),\ \vec{n}=(1,0,-1)$ $\vec{n}\cdot\overrightarrow{OA}=1, \quad |\vec{n}|=\sqrt2$ $d(H_0)=\dfrac{1}{\sqrt2} \Rightarrow \text{(P)}\to\text{(5)}$ $\text{dist}\big((0,1,2),H_0\big)=\dfrac{|0-2|}{\sqrt2}$ $=\dfrac{2}{\sqrt2}=\sqrt2 \Rightarrow \text{(Q)}\to\text{(4)}$ $O(0,0,0)\in H_0 \Rightarrow \text{dist}(O,H_0)=0$ $\Rightarrow \text{(R)}\to\text{(3)}$ $y=z,\ x=1,\ x=z \Rightarrow (1,1,1)$ $\text{dist from } O=\sqrt{1+1+1}=\sqrt3 \Rightarrow \text{(S)}\to\text{(1)}$ $\text{(P)}\to(5),\ \text{(Q)}\to(4),\ \text{(R)}\to(3),\ \text{(S)}\to(1)$ $\boxed{\text{Answer: Option (2)}}$