List of top Mathematics Questions on 3D Geometry asked in JEE Main

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

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

Let a line \( l \) pass through the origin and be perpendicular to the lines \[ l_1: \vec{r}_1 = i + j + 7k + \lambda(i + 2j + 3k), \quad \lambda \in \mathbb{R} \] \[ l_2: \vec{r}_2 = -i + j + 2k + \mu(i + 2j + k), \quad \mu \in \mathbb{R} \] If \( P \) is the point of intersection of \( l_1 \) and \( l_2 \), and \( Q (a, b, \gamma) \) is the foot of perpendicular from P on \( l \), then \( (a + b + \gamma) \) is equal to:

Let (α, β, γ) be the image of the point P(3, 3, 5) in the plane 2x + y - 3z = 6. Then α + β + γ is equal to:

If equation of the plane that contains the point \((-2,3,5)\) and is perpendicular to each of the planes \( 2x + 4y + 5z = 8 \) and \( 3x - 2y + 3z = 5 \), is \( \alpha x + \beta y + \gamma z = 97 \), then \( \alpha + \beta + \gamma \) is:

The distance of the point $(1, -2, 3)$ from the plane $x - y + z = 5$ measured parallel to a line, whose direction ratios are $2, 3, -6$ is :

Suppose the line $\frac{x - 2}{\alpha} = \frac{y - 2}{-5} = \frac{z + 2}{2}$ lies on the plane $x + 3y - 2z + \beta = 0$. Then $(\alpha + \beta)$ is equal to _________.

Let P be a plane containing the line (x-1)/3 = (y+6)/4 = (z+5)/2 and parallel to the line (x-3)/4 = (y-2)/(-3) = (z+5)/7. If the point (1, -1, α) lies on the plane P, then the value of |α| is equal to ________.

Let the mirror image of the point (1, 3, a) with respect to the plane r · (2 i - j + k) - b = 0 be (-3, 5, 2). Then, the value of |a + b| is equal to ________.

Let the plane ax + by + cz + d = 0 bisect the line joining the points (4, -3, 1) and (2, 3, -5) at the right angles. If a, b, c, d are integers, then the minimum value of (a² + b² + c² + d²) is __________.

The equation of the planes parallel to the plane x - 2y + 2z - 3 = 0 which are at unit distance from the point (1, 2, 3) is ax + by + cz + d = 0. If (b - d) = K(c - a), then the positive value of K is __________.

If the shortest distance between x - λ = 2y - 1 = -2z and x = y + 2λ = z - λ is √7 / 2√2, then |λ| is _______

Let line L pass through the point \( (0, 1, 2) \), intersect the line \( {2 = {3 = {4 \), and be parallel to the plane \( 2x + y - 3z = 4 \). Find the distance of the point \( P(1, -9, 2) \) from line L. \\

Let the vectors \( , , \) represent three coterminous edges of a parallelepiped of volume \( V \). Then the volume of the parallelepiped, whose coterminous edges are represented by \( , + \) and \( + 2 + 3 \) is equal to: \\ \\