3D Geometry Questions (578)

A perpendicular is drawn from a point on the line \(\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z}{1}\) to the plane \(x + y + z = 3\) such that the foot of the perpendicular \(Q\) also lies on the plane \(x - y + z = 3\). The co-ordinates of \(Q\) are:

Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is

The minimum value of $x^2+y^2+z^2$ if $ax+by+cz=p$ is

If direction cosines of a line \(L\) satisfy \(l = -m - n\) and \(l^2 = m^2 + n^2\), then the angle \(\theta\) that the line makes is such that \(\cos\theta =\):

If the angle between the line \(2(x+1) = y = z + 4\) and the plane \(2x - y + \sqrt{\lambda}z + 4 = 0\) is \(\pi/6\), then the value of \(\lambda\) is

The distance of the point \((1, 0, 2)\) from the point of intersection of the line \(\dfrac{x-2}{3} = \dfrac{y+1}{4} = \dfrac{z-2}{12}\) and the plane \(x - y + z = 16\), is

Two systems of rectangular axes have the same origin. If a plane cuts them at distances \(a, b, c\) and \(a', b', c'\) from the origin, then

Line x = ay + b, z = cy + d and line x = a'z + b', y = c' + d' are perpendicular to each other. Then which condition holds?

Given two planes \(P_1: 2x - y - 4 = 0\) and \(P_2: y + 2z - 4 = 0\) and point \(K(1, 1, 0)\). Let a third plane \(P_3\) pass through \(K\) and satisfy \(P_3: P_1 + \lambda P_2 = 0\). The equation of plane \(P_3\) is:

ABC is a triangle with vertices A(0, 0, 6), B(0, 4, 0) and C(6, 0, 0). Let points D, E and F are the mid-points of BC, AC and AB, respectively. Find the length of median AD.

Two lines \(\frac{x - 3}{1} = \frac{y + 1}{3} = \frac{z - 6}{-1}\) and \(\frac{x + 5}{7} = \frac{y - 2}{-6} = \frac{z - 3}{4}\) intersect at the point R. The reflection of R in the xy-plane has coordinates

Let (l, 2, 1) be a point on the plane which passes through the point (4, -2, 2). If the plane is perpendicular to the line joining the points (-2, -21, 29) and (-1, -16, 23), then \(\left(\frac{l}{11}\right)^2 - \frac{4l}{11} - 4\) is equal to

The equation of the plane with intercepts 2, 3 and 4 on the X, Y and Z-axes respectively, is

The equation of the plane passing through the point (0, 7, -7) and containing the line \(\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z+2}{1}\) is

The direction ratios of normal to the plane through the points (0, −1, 0) and (0, 0, 1) and making an angle π/4 with the plane y − z + 5 = 0 are

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:

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 the planes $x = cy + bz, y = az + cx, z = bx + ay$ pass through one line then the value of $a^2 + b^2 + c^2 + 2abc$ is __________.

A variable plane is at a constant distance $p$ from the origin and meets the axes at $A, B, C$. If the locus of the centroid of the tetrahedron $OABC$ is $x^{-2} + y^{-2} + z^{-2} = 2qp^{-2}$ then the value of $\sqrt{\lambda}$ is __________.

If the planes $x - y + z + 1 = 0, \lambda x + 3y + 2z - 3 = 0, 3x + \lambda y + z - 2 = 0$ form a triangular prism then $\lambda$ is __________.

If the distance between the plane $Ax - 2y + z = d$ and the plane containing the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{3} = \frac{y-3}{4} = \frac{z-4}{5}$ is $\sqrt{6}$, then $|d|$ is equal to __________.

Let \(A(2, 3, 5)\), \(B(-1, 3, 2)\) and \(C(\lambda, 5, \mu)\) be the vertices of a \(\triangle ABC\). If the median through \(A\) is equally inclined to the coordinate axes, then

Find the image of point A(2, 1, 6) about the mirror plane x + y - 2z = 3.

Ex. 45 Statement I: Line \(\frac{x-1}{3} = \frac{y-2}{11} = \frac{z+1}{11}\) lies in the plane \(11x - 3z - 14 = 0\).Statement II: A straight line lies in a plane, if the line is parallel to the plane and a point of the line is in the plane.

Given the planes x + 2y − 3z + 2 = 0 and x − 2y + 3z + 7 = 0, if the point P is (1, 2, 2), then

For positive l, m and n, if the planes x = ny + mz, y = lz + nx, z = mz + ly intersect in a straight line, then l, m and n satisfy the equation

Through a point \(P(h, k, l)\) a plane is drawn at right angles to OP to meet the coordinate axes in A, B and C. If \(OP = p\), \(A_{xy}\) is area of projection of \(\triangle ABC\) on xy-plane, \(A_{yz}\) is area of projection of \(\triangle ABC\) on yz-plane, then \(\frac{A_{xy}}{A_{yz}}\)

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:

$P$ is a point on the plane $lx + my + nz = p$. A point $Q$ is taken on the line $OP$ such that $OP.OQ = p^2$. Then the locus of $Q$ is:

The orthogonal projection of the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{4}$ on the plane $3x + 4y + 5z = 0$ is:

If $abc \neq 0$ and let $(p_1, q_1, r_1)$ be the image of $(p, q, r)$ in the plane $ax + by + cz + d = 0$, then:

The foot of the perpendicular from the point $O(0, 0, 0)$ to the line of intersection of the planes $x + y + z = 4$ and $2x + y + 3z = 1$ is point $A$. Then the equation of line $OA$ is:

A variable line passing through the point $P(0, 0, 2)$ always makes angle $60°$ with $z$-axis, intersects the plane $x + y + z = 1$. Then the locus of point of intersection of the line and the plane is:

The locus of intersection of locus of $P$ with the plane $x + y + z = 1$ is:

A mirror and a source of light are situated at the origin $O$ and at a point on the line $OX$ respectively. A ray of light from the source strikes the mirror at $O$ and is reflected. If the direction ratios of the normal to the plane of the mirror are $(1, -1, 1)$; then the direction cosines of the reflected ray are :

Given \(\overrightarrow{OQ} = (1-3\mu)\hat{i} + (\mu-1)\hat{j} + (5\mu+2)\hat{k}\) and \(\overrightarrow{OP} = 3\hat{i} + 2\hat{j} + 6\hat{k}\) (where O is the origin). If \(\overrightarrow{PQ}\) is parallel to the plane \(x - 4y + 3z = 1\), find the value of \(\mu\).

The line $\frac{x - 2}{3} = \frac{y + 1}{2} = \frac{z - 1}{-1}$ intersects the curve $xy = c^2, z = 0$ if $c$ is equal to:

If the line $x = y = z$ intersect the line in $A x + \sin B y + \sin C z = 2d^2$, sin $2A x + \sin 2B y + \sin 2C z = d^2$ then $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ is equal to : (where $A + B + C = \pi$)

The equation of a line in $xz$ plane equally inclined with $x$ and $z$ axes which is at a unit distance from the line $\frac{x-1}{1} = \frac{y-1}{1} = \frac{z}{1}$ is:

The coordinate of the points on the line $\frac{x+2}{3} = \frac{y+1}{2} = \frac{z-3}{2}$ which are at a distance $3\sqrt{2}$ from the point $(1, 2, 3)$

The equation of the plane parallel to plane $x + y + 2z = 5$ at a distance $\sqrt{6}$ units from the plane is/are:

The shortest distance between the lines $2x + y + z = 1, 3x + y + 2z = 2$ and $x + y = z$ is $d$ then $\frac{1}{d^2} = $ ______.

Three lines $y - z - 1 = 0, x = 0; x + z - 1 = 0, y = 0; x - z - 1 = 0, y = 0$ intersect the $xy$ plane at $A, B$ and $C$. If the orthocentre of $\triangle ABC$ is $(p, q, r)$ then $3p + q + r = $ __________.

The minimum distance of the point $(1, 1, 1)$ from the plane $x + y + z = 1$ measured perpendicular to the line $\frac{x-x_1}{l} = \frac{y-y_1}{2} = \frac{z-z_1}{3}$ is then $\frac{3\ell^2}{\gamma} = $ __________.

The maximum distance between the point $P(0, 0, 3)$ and the circle $x^2 + y^2 - 2\sqrt{5}x - 4y + 8 = 0; z = 0$ is __________.

Plane $2x + 3y + 4z = 5$ is rotated about the line where it cuts the $xy$-plane by an angle $\alpha$. In the new position the plane contains the point $(1, 1, 1)$. If the angle $\alpha = \cos^{-1}\sqrt{\frac{p}{q}}, (p$ is rational number in its simplest form$)$ then $q - 2p = $ __________.

If $a, b, c$ be the lengths of the intercepts of the plane passing through the intersection of the planes $2x + y + 2z = 9, 4x - 5y - 4z = 1$ and the point $(3, 2, 1)$ on the coordinate axes, then $(5a + b + c)/2 = $ __________.

If $Q$ is the foot of perpendicular from the point $P(4, -5, 3)$ on the line $\frac{x-5}{3} = \frac{y+2}{-4} = \frac{z-6}{5}$, then $[PQ] = $ __________. (where $[.]$ denote greatest integer function)

The projection of the line $\frac{x}{5} = \frac{y-1}{2} = \frac{z-1}{1}$ on a plane $P$ is $\frac{x}{1} = \frac{y-1}{1} = \frac{z-1}{-1}$. The plane $P$ passes through $(k, -2, 0)$ then $k = $ __________.

Let for $\lambda \in [0, \infty)$ such that $(x, y, z) \neq (0, 0, 0)$ and $(\vec{i} + \vec{j} + 3\vec{k})x + (3\vec{i} - \vec{j} + \vec{k})y + (4\vec{i} + 5\vec{j})z = \lambda(x\vec{i} + y\vec{j} + 3\vec{k})$, then the value of $\frac{x - y - z}{x}$ is equal to __________.