Skew lines

The shortest distance between the lines $\vec r=\left(\dfrac13\hat i+2\hat j+\dfrac83\hat k\right)+\lambda\left(2\hat i-5\hat j+6\hat k\right)$ and $\vec r=\left(-\dfrac23\hat i-\dfrac13\hat k\right)+\mu\left(\hat j-\hat k\right)$, $\lambda,\mu\in\mathbb R$, is:

The shortest distance between the lines $\dfrac{x-4}1=\dfrac{y-3}2=\dfrac{z-2}{-3}$ and $\dfrac{x+2}2=\dfrac{y-6}4=\dfrac{z-5}{-5}$ is :

Let $Q$ be the cube with the set of vertices $\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \in \{0, 1\}\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_1$ and $\ell_2$, let $d(\ell_1, \ell_2)$ denote the shortest distance between them. Then the maximum value of $d(\ell_1, \ell_2)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is

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.

The correct option is:

Let $P_1$ and $P_2$ be two planes given by
$$P_1:\ 10x + 15y + 12z - 60 = 0, \qquad P_2:\ -2x + 5y + 4z - 20 = 0.$$
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_1$ and $P_2$?

If the shortest distance between the lines $\vec r=\left(-\hat i+3\hat k\right)+\lambda\left(\hat i-a\hat j\right)$ and $\vec r=\left(-\hat j+2\hat k\right)+\mu\left(\hat i-\hat j+\hat k\right)$ is $\sqrt{\dfrac23}$, then the integral value of $a$ is equal to ______.

If the shortest distance between the lines $\dfrac{x-1}2=\dfrac{y-2}3=\dfrac{z-3}\lambda$ and $\dfrac{x-2}1=\dfrac{y-4}4=\dfrac{z-5}5$ is $\dfrac1{\sqrt3}$, then the sum of all possible values of $\lambda$ is:

Let $l_1$ be the line in $xy$-plane with $x$ and $y$ intercepts $\dfrac18$ and $\dfrac1{4\sqrt2}$ respectively, and $l_2$ be the line in $zx$-plane with $x$ and $z$ intercepts $-\dfrac18$ and $-\dfrac1{6\sqrt3}$ respectively. If $d$ is the shortest distance between the line $l_1$ and $l_2$, then $d^{-2}$ is equal to ______.

If the lines $\vec r=\left(\hat i-\hat j+\hat k\right)+\lambda\left(3\hat j-\hat k\right)$ and $\vec r=\left(\alpha\hat i-\hat j\right)+\mu\left(2\hat i-3\hat k\right)$ are co-planar, then the distance of the plane containing these two lines from the point $(\alpha,0,0)$ is:

The shortest distance between the lines $\dfrac{x-3}2=\dfrac{y-2}3=\dfrac{z-1}{-1}$ and $\dfrac{x+3}2=\dfrac{y-6}1=\dfrac{z-5}3$, is

The line of shortest distance between the lines $\dfrac{x-2}0=\dfrac{y-1}1=\dfrac z1$ and $\dfrac{x-3}2=\dfrac{y-5}2=\dfrac{z-1}1$ makes an angle of $\cos^{-1}\left(\sqrt{\dfrac2{27}}\right)$ with the plane $P:ax-y-z=0$, $(a>0)$. If the image of the point $(1,1,-5)$ in the plane P is $(\alpha,\beta,\gamma)$, then $\alpha+\beta-\gamma$ is equal to ______.

The shortest distance between the lines $\dfrac{x+7}{-6}=\dfrac{y-6}7=z$ and $\dfrac{7-x}2=y-2=z-6$ is

Let the lines $\dfrac{x-1}\lambda=\dfrac{y-2}1=\dfrac{z-3}2$ and $\dfrac{x+26}{-2}=\dfrac{y+18}3=\dfrac{z+28}\lambda$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?

Let L$_1$ be a line in the yz - plane with y and z intercepts $\dfrac14$ and $\dfrac1C$ (C > 0) respectively. Let L$_2$ be the line in the xz - plane with x and z intercepts $\dfrac13$ and $-\dfrac1C$ respectively. If the shortest distance between L$_1$ and L$_2$ is $\dfrac15$, then C$^2$ is equal to _______.

Let $L_1$ and $L_2$ denote the lines

$$\vec r = \hat i + \lambda(-\hat i + 2\hat j + 2\hat k),\ \lambda\in\mathbb R\quad\text{and}\quad\vec r = \mu(2\hat i - \hat j + 2\hat k),\ \mu\in\mathbb R$$

respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$?

If the line, $\dfrac{x-3}1=\dfrac{y+2}{-1}=\dfrac{z+\lambda}{-2}$ lies in the plane, $2x-4y+3z=2$, then the shortest distance between this line and the line, $\dfrac{x-1}{12}=\dfrac y9=\dfrac z4$ is :

The shortest distance between the lines $\dfrac x2=\dfrac y2=\dfrac z1$ and $\dfrac{x+2}{-1}=\dfrac{y-4}8=\dfrac{z-5}4$ lies in the interval :

The number of distinct real values of λ for which the lines $\dfrac{x-1}1=\dfrac{y-2}2=\dfrac{z+3}{\lambda^2}$ and $\dfrac{x-3}1=\dfrac{y-2}{\lambda^2}=\dfrac{z-1}2$ are coplanar is :

If the shortest distance between the lines $\dfrac{x-1}\alpha=\dfrac{y+1}{-1}=\dfrac z1$, $(\alpha\ne-1)$ and $x+y+z+1=0=2x-y+z+3$ is $\dfrac1{\sqrt3}$, then a value of α is :

The shortest distance between the $z$-axis and the line $x+y+2z-3=0=2x+3y+4z-4$, is :