JEE Advanced 2023 Paper 1Question 1
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
(P) The value of $d(H_0)$ is
(Q) The distance of the point $(0,1,2)$ from $H_0$ is
(R) The distance of origin from $H_0$ is
(S) The distance of origin from the point of intersection of planes $y=z$, $x=1$ and $H_0$ is
List-II
(1) $\sqrt3$
(2) $\frac{1}{\sqrt3}$
(3) 0
(4) $\sqrt2$
(5) $\frac{1}{\sqrt2}$
The correct option is:
3D Geometry
Solution
For finite $d(H)$, $H\parallel\ell_2$, so $n\perp(1,1,1),(1,0,1)$: $n=(1,0,-1)$, $H_0: x=z$. Then $d(H_0)=\frac1{\sqrt2}$; distance of $(0,1,2)$ is $\sqrt2$; of origin $0$; intersection with $y=z,x=1$ is $(1,1,1)$, distance $\sqrt3$.