Question

Let l₁ and l₂ be the lines r₁ = λ(\(\hat{i}+\hat{j}+\hat{k}\)) and r₂ = (\(\hat{j}-\hat{k}\)) + μ (\(\hat{i}+\hat{k}\)), respectively. Let X be the set of all the planes H containing line l₁. For a plane H, let d (H) denote the smallest possible distance between the points of l₂ 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
(P)	The value of d (H0) is	(1)	\(\sqrt3\)
(Q)	The distance of the point (0,1,2) from H0 is	(2)	\(\frac{1}{\sqrt3}\)
(R)	The distance of origin from H0 is	(3)	0
(S)	The distance of origin from the point of intersection of planes y = z, x = 1, and H0 is	(4)	\(\sqrt2\)
		(5)	\(\frac{1}{\sqrt2}\)

 The correct option is: 

• A. (P)\(\rightarrow\)(2) (Q)\(\rightarrow\)(4) (R)\(\rightarrow\)(5) (S)\(\rightarrow\)(1)
• B. (P)\(\rightarrow\)(5) (Q)\(\rightarrow\)(4) (R)\(\rightarrow\)(3) (S)\(\rightarrow\)(1)
• C. (P)\(\rightarrow \)(2) (Q)\(\rightarrow\)(1) (R)\(\rightarrow\)(3) (S)\(\rightarrow\)(2)
• D. (P)\(\rightarrow\)(5) (Q)\(\rightarrow\)(1) (R)\(\rightarrow\)(4) (S)\(\rightarrow\)(2)

Answer 1

The Correct Option is B

Plane and Normal Vector Calculation

The correct option is (B).

Normal Vector of Plane Parallel \( l_1 \) and \( l_2 \):

The normal vector of the plane parallel to lines \( l_1 \) and \( l_2 \) is:

\[ \begin{vmatrix} i & j & k \\ 1 & 1 & 1 \\ 1 & 0 & 1 \end{vmatrix} = \hat{j}(1) - \hat{j}(1-1) + \hat{k}(-1) \]

\[ = \hat{i} - \hat{j} \]

Equation of Plane \( H_0 \): 

\[ H_0 : x - z = c \quad \text{where} \quad (0, 0, 0) \]

\[ \Rightarrow c = 0 \]

\[ H_0 : x - z = 0 \]

Distance from the Point (0, 1, -1) to the Plane:

(P) The distance of point (0, 1, -1) from \( H_0 \) is:

\[ d(H_0) = \left| \frac{0 - (-1)}{\sqrt{2}} \right| = \frac{1}{\sqrt{2}} \quad \Rightarrow \quad P \rightarrow 5 \]

(Q) The distance from the point (0, 2, 0) to the plane is:

\[ d = \left| \frac{0 - 2}{\sqrt{2}} \right| = \sqrt{2} \quad \Rightarrow \quad Q \rightarrow 4 \]

(R) The distance from the origin to the plane is:

\[ d = \left| \frac{0}{\sqrt{2}} \right| = 0 \quad \Rightarrow \quad S \rightarrow 3 \]

Intersection Point and Distance:

(S) The point of intersection is (1, 1, 1):

\[ \Rightarrow S \rightarrow 1 \]

Distance is:

\[ d = \sqrt{1 + 1 + 1} = \sqrt{3} \]