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If the shortest distance between the lines \( \dfrac{x - 4}{1} = \dfrac{y + 1}{2} = \dfrac{z}{-3} \) and \( \dfrac{x - \lambda}{2} = \dfrac{y + 1}{4} = \dfrac{z - 2}{-5} \) is \( \dfrac{6}{\sqrt{5}} \), then the sum of all possible values of \( \lambda \) is:

A line with direction ratios \( 2,1,2 \) meets the lines \( x=y+2=z \) and \( x+2=2y=2z \) respectively at the points \( P \) and \( Q \). If the length of the perpendicular from the point \( (1,2,12) \) to the line \( PQ \) is \( l \), then \( l^{2} \) is __________.

If \( A(1,-1,2), \ B(5,7,-6), \ C(3,4,-10) \) and \( D(-1, -4, -2) \) are the vertices of a quadrilateral \( ABCD \), then its area is:

A line passes through \( A(4,-6,-2) \) and \( B(16,-2,4) \). The point \( P(a,b,c) \), where \( a,b,c \) are non‑negative integers, on the line \( AB \) lies at a distance of \( 21 \) units from the point \( A \). The distance between the points \( P(a,b,c) \) and \( Q(4,-12,3) \) is equal to ________.

If the shortest distance between the lines \( \frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1} \) and \( \frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1} \) is \( 1 \), then the sum of all possible values of \( \lambda \) is:

Let \(d\) be the distance of the point of intersection of the lines \( \dfrac{x+6}{3}=\dfrac{y}{2}=\dfrac{z+1}{1} \) and \( \dfrac{x-7}{4}=\dfrac{y-9}{3}=\dfrac{z-4}{2} \) from the point \( (7,8,9) \). Then \( d^{2}+6 \) is equal to:

If the shortest distance between the lines;

\( L_{1} : \ \vec{r} = (2 + \lambda)\hat{i} + (1 - 3\lambda)\hat{j} + (3 + 4\lambda)\hat{k}, \ \lambda \in \mathbb{R} \)

\( L_{2} : \ \vec{r} = 2(1 + \mu)\hat{i} + 3(1 + \mu)\hat{j} + (5 + \mu)\hat{k}, \ \mu \in \mathbb{R} \)

is \( \dfrac{m}{\sqrt{n}} \), where \( \gcd(m, n) = 1 \), then the value of \( m + n \) equals:

If the shortest distance between the lines \( \frac{x-\lambda}{2}=\frac{y-4}{4}=\frac{z-3}{4} \) and \( \frac{x-2}{4}=\frac{y-4}{6}=\frac{z-7}{8} \) is \( \frac{13}{\sqrt{29}} \), then a value of \( \lambda \) is:

If the shortest distance between the lines \( \frac{x+2}{2} = \frac{y+3}{3} = \frac{z-5}{4} \) and \( \frac{x-3}{1} = \frac{y-2}{-3} = \frac{z+4}{2} \) is \( \frac{38}{3\sqrt{5}} \, k \), and \( \int_{0}^{k} [ x^{2} ]dx = \alpha - \sqrt{\alpha} \), where \( [x] \) denotes the greatest integer function, then \( 6\alpha^{3} \) is equal to ________.

If the mirror image of the point \( P(3, 4, 9) \) in the line \( \dfrac{x-1}{3} = \dfrac{y+1}{2} = \dfrac{z-2}{1} \) is \( (\alpha,\beta,\gamma) \), then \( 14(\alpha + \beta + \gamma) \) is:

If the line \( \frac{2 - x}{3} = \frac{3y - 2}{4\lambda + 1} = 4 - z \) makes a right angle with the line \( \frac{x + 3}{3\mu} = \frac{1 - 2y}{6} = \frac{5 - z}{7} \), then \( 4\lambda + 9\mu \) is equal to:

Let the point, on the line passing through the points \( P(1, -2, 3) \) and \( Q(5, -4, 7) \), farther from the origin and at a distance of \( 9 \) units from the point \( P \), be \( (\alpha, \beta, \gamma) \). Then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:

The shortest distance between the lines \( \frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5} \) and \( \frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1} \) is:

Let \( L_{1}:\ \vec{r} = (\hat{\imath}-\hat{\jmath}+2\hat{k}) + \lambda(\hat{\imath}-\hat{\jmath}+2\hat{k}),\ \lambda \in \mathbb{R}; \ L_{2}:\ \vec{r} = (j-\hat{k}\,) + \mu(3\hat{\imath}+\hat{\jmath}+p\hat{k}),\ \mu \in \mathbb{R}; \) and \( L_{3}:\ \vec{r} = \delta(\hat{\imath}+m\hat{\jmath}+n\hat{k}),\ \delta \in \mathbb{R} \) be three lines such that \( L_{1} \) is perpendicular to \( L_{2} \) and \( L_{3} \) is perpendicular to both \( L_{1} \) and \( L_{2} \). Then, the point which lies on \( L_{3} \) is:

The distance of the point $Q(0, 2, -2)$ from the line passing through the point $P(5, -4, 3)$ and perpendicular to the lines $\vec{r} = (-3\hat{i} + 2\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 5\hat{k}), \lambda \in R$ and $\vec{r} = (\hat{i} - 2\hat{j} + \hat{k}) + \mu(-\hat{i} + 3\hat{j} + 2\hat{k}), \mu \in R$ is:

Let \( P(x, y, z) \) be a point in the first octant, whose projection in the \( xy \)-plane is the point \( Q \). Let \( OP = \gamma \); the angle between \( OQ \) and the positive \( x \)-axis be \( \theta \); and the angle between \( OP \) and the positive \( z \)-axis be \( \phi \), where \( O \) is the origin. Then the distance of \( P \) from the \( x \)-axis is:

The shortest distance between the lines \( \frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5} \) and \( \frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3} \) is:

The lines \( \dfrac{x-2}{2}=\dfrac{y}{-2}=\dfrac{z-7}{16} \) and \( \dfrac{x+3}{4}=\dfrac{y+2}{3}=\dfrac{z+2}{1} \) intersect at the point \( P \). If the distance of \( P \) from the line \( \dfrac{x+1}{2}=\dfrac{y-1}{3}=\dfrac{z-1}{1} \) is \( l \), then \( 14\, l^{2} \) is equal to __________ .

Let \( P \) be the point \( (10, -2, -1) \) and \( Q \) be the foot of the perpendicular drawn from the point \( R(1, 7, 6) \) on the line passing through the points \( (2, -5, 11) \) and \( (-6, 7, -5) \). Then the length of the line segment \( PQ \) is equal to:

Let the line \( L \) intersect the lines \( x - 2 = -y = z - 1 \), \( 2(x + 1) = 2(y - 1) = z + 1 \) and be parallel to the line \( \dfrac{x - 2}{3} = \dfrac{y - 1}{1} = \dfrac{z - 2}{2} \). Then which of the following points lies on \( L \)?

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \) and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is:

The square of the distance of the image of the point \( (6,1,5) \) in the line \( \dfrac{x-1}{3}=\dfrac{y}{2}=\dfrac{z-2}{4} \), from the origin is_________.

Let \( (\alpha,\beta,\gamma) \) be the mirror image of the point \( (2,3,5) \) in the line \( \dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4} \). Then \( 2\alpha+3\beta+4\gamma \) is equal to:

If \( d_1 \) is the shortest distance between the lines \( x + 1 = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) and \( d_2 \) is the shortest distance between the lines \( \dfrac{x - 1}{2} = \dfrac{y + 8}{-7} = \dfrac{z - 4}{5} \) and \( \dfrac{x - 1}{2} = \dfrac{y - 2}{1} = \dfrac{z - 6}{-3} \), then the value of \( \dfrac{32\sqrt{3}\, d_1}{d_2} \) is:

Let the image of the point \( (1,0,7) \) in the line \(\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \) be the point \( (\alpha,\beta,\gamma) \). Then which one of the following points lies on the line passing through \( (\alpha,\beta,\gamma) \) and making angles \(\displaystyle \frac{2\pi}{3} \) and \(\displaystyle \frac{3\pi}{4} \) with \( y \)-axis and \( z \)-axis respectively and an acute angle with \( x \)-axis?

Let the point \( (+1, \alpha, \beta) \) lie on the line of the shortest distance between the lines \( \dfrac{x + 2}{-3} = \dfrac{y - 2}{4} = \dfrac{z - 5}{2} \) and \( \dfrac{x + 2}{1} = \dfrac{y + 6}{2} = \dfrac{z - 1}{0} \).Then \( (\alpha - \beta)^{2} \) is equal to \( \_\_\_\_\_\_ \).

Let \( (\alpha,\beta,\gamma) \) be the image of the point \( (8,5,7) \) in the line: \( \dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{5} \). Then \( \alpha+\beta+\gamma \) is equal to:

Let \( Q \) and \( R \) be the feet of perpendiculars from the point \( P(a,a,a) \) on the lines \( x=y, \ z=1 \) and \( x=-y, \ z=-1 \) respectively. If \( \angle QPR \) is a right angle, then \( 12a^2 \) is equal to:

Let \( P \) be the point of intersection of the lines \( \dfrac{x-2}{1} = \dfrac{y-4}{5} = \dfrac{z-2}{1} \) and \( \dfrac{x-3}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{2} \). Then, the shortest distance of \( P \) from the line \( 4x = 2y = z \) is:

Consider a line \( L \) passing through the points \( P(1,2,1) \) and \( Q(2,1,-1) \). If the mirror image of the point \( A(2,2,2) \) in the line \( L \) is \( (\alpha,\beta,\gamma) \), then \( \alpha+\beta+6\gamma \) is equal to ________________.

Let \( P(\alpha,\beta,\gamma) \) be the image of the point \( Q(1,6,4) \) in the line \(\displaystyle \frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3} \). Then \( 2\alpha+\beta+\gamma \) equal to ________.

Let \( P \) and \( Q \) be the points on the line \( \dfrac{x+3}{8} = \dfrac{y-4}{2} = \dfrac{z+1}{2} \) which are at a distance of \( 6 \) units from the point \( R(1, 2, 3) \). If the centroid of the triangle \( PQR \) is \( (\alpha,\beta,\gamma) \), then \( \alpha^{2} + \beta^{2} + \gamma^{2} \) is:

Let \( O \) be the origin, and \( M \) and \( N \) be the points on the lines \( \dfrac{x-5}{4}=\dfrac{y-4}{1}=\dfrac{z-5}{3} \) and \( \dfrac{x+8}{12}=\dfrac{y+2}{5}=\dfrac{z-11}{9} \) respectively such that \( MN \) is the shortest distance between the given lines. Then \( \overrightarrow{OM}\cdot\overrightarrow{ON} \) is equal to ________.

Consider the line \( L \) passing through the points \( (1, 2, 3) \) and \( (2, 3, 5) \). The distance of the point \(\displaystyle \left( \frac{11}{3}, \frac{11}{3}, \frac{19}{3} \right) \) from the line \( L \) along the line \(\displaystyle \frac{3x-11}{2} = \frac{3y-11}{1} = \frac{3z-19}{2} \) is equal to

Let a line passing through the point \( (-1, 2, 3) \) intersects the lines \( L_1 : \dfrac{x-1}{3}=\dfrac{y-2}{2}=\dfrac{z+1}{-2} \) at \( M(\alpha,\beta,\gamma) \) and \( L_2 : \dfrac{x+2}{-3}=\dfrac{y-2}{-2}=\dfrac{z-1}{4} \) at \( N(a,b,c) \). Then, the value of \( \dfrac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2} \) equals _______.

Let the line of the shortest distance between the lines \( L_1 : \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}) \) and \( L_2 : \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(\hat{i} + \hat{j} - \hat{k}) \) intersect \( L_1 \) and \( L_2 \) at \( P \) and \( Q \) respectively. If \( (\alpha, \beta, \gamma) \) is the mid point of the line segment \( PQ \), then \( 2(\alpha + \beta + \gamma) \) is equal to ______.

The shortest distance, between lines \( L_1 \) and \( L_2 \), where \( L_1:\; \dfrac{x-1}{2}=\dfrac{y+1}{-3}=\dfrac{z+4}{2} \) and \( L_2 \) s the line, passing through the line, passing through the points \( A=(-4,4,3), B=(-1,6,3) \) and perpendicular to the line \( \dfrac{x-3}{-2}=\dfrac{y}{3}=\dfrac{z-1}{1} \) is:

Let \( PQR \) be a triangle with \( R(-1, 4, 2) \). Suppose \( M(2, 1, 2) \) is the mid point of \( PQ \). The distance of the centroid of \( \Delta PQR \) from the point of intersection of the lines \( \frac{x-2}{0} = \frac{y}{2} = \frac{z+3}{-1} \) and \( \frac{x-1}{1} = \frac{y+3}{-3} = \frac{z+1}{1} \) is:

The distance, of the point \( (7, -2, 11) \) from the line \( \dfrac{x-6}{1} = \dfrac{y-4}{0} = \dfrac{z-8}{3} \) along the line \( \dfrac{x-5}{2} = \dfrac{y-1}{-3} = \dfrac{z-5}{6} \), is:

Let \( (\alpha, \beta, \gamma) \) be the foot of perpendicular from the point \( (1, 2, 3) \) on the line \( \frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} \). Then \( 19(\alpha + \beta + \gamma) \) is equal to:

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JEE Main 2024 29th Jan Evening Shift

Let \( P(3, 2, 3) \), \( Q(4, 6, 2) \) and \( R(7, 3, 2) \) be the vertices of \( \triangle PQR \). Then, the angle \( \angle QPR \) is: