3D Geometry Questions - JEE

If $P(-1, 2, -3)$ and $Q(3, 0, 3)$ are two points on the plane $P_1: 2x + y - z = 3$ and $R(x_0, y_0, z_0)$ be a point such that $x_0 - 2y_0 + 3z_0 + 1 = 0$ and $|PR - QR|$ is maximum, then $(x_0 + y_0 + z_0)$ is equal to:

Ex. 61 (D): The angle between the planes $x + y + z = 0$ and $3x - 4y + 5z = 0$ is

A line makes the same angle $\theta$ with each of the $x$ and $z$ axis. If the angle $\beta$, which it makes with $y$-axis, is such that $\sin^2\beta = 3\sin^2\theta$, then $\cos^2\theta$ equals

A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is

The length of the perpendicular from the origin to the plane passing through the point $\vec{a}$ and containing the line $\vec{r} = \vec{b} + \lambda \vec{c}$ is

Consider a plane p: $\vec{r} \cdot \vec{n} = d$ (where $\vec{n}$ is not a unit vector). There are two points A($\vec{a}$) and B($\vec{b}$) lying on the same side of the plane. If foot of perpendicular from A and B to the plane p are P and Q respectively, then length of PQ is:

The equation of the plane through the intersection of the planes $x + y + z = 1$ and $2x + 3y - z + 4 = 0$ and parallel to X-axis, is

The distance of the point $(1, -5, 9)$ from the plane $x - y + z = 5$ measured along the line $x = y = z$ is

Point of intersection of the line lies on:

Let line $L_1$ be parallel to $-3\hat{i}+2\hat{j}+4\hat{k}$ and pass through $(2,6,7)$, and $L_2$ be parallel to $2\hat{i}+\hat{j}+3\hat{k}$ passing through $(4,3,5)$. If $L_3$ is parallel to $-3\hat{i}+5\hat{j}+16\hat{k}$ and intersects $L_1$ and $L_2$ at points $C$ and $D$, then $|\overrightarrow{CD}|^2$ is equal to:

If $P(-1, 2, -3)$ and $Q(3, 0, 3)$ are two points on the plane $P_1: 2x + y - z = 3$ and $R(x_0, y_0, z_0)$ be a point such that $x_0 - 2y_0 + 3z_0 + 1 = 0$ and $|PR - QR|$ is maximum, then $(x_0 + y_0 + z_0)$ is equal to:

The shortest distance from the plane $12x + 4y + 3z = 327$ to the sphere $x^2 + y^2 + z^2 + 4x - 2y - 6z = 155$ is

The points $(5, -4, 2)$, $(4, -3, 1)$, $(7, -6, 4)$ and $(8, -7, 5)$ are the vertices of:

If the line $\dfrac{x-2}{3} = \dfrac{y+1}{2} = \dfrac{z-1}{-1}$ intersects the plane $2x + 3y - z + 13 = 0$ at a point $P$ and the plane $3x + y + 4z = 16$ at a point $Q$, then $PQ$ is equal to ______ (up to three decimal places).

Ex. 61 (C): The angle between the line $x = y = z$ and the plane $4x - 3y + 5z = 2$ is

The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines $\vec{r} = (\hat{i} + \hat{j}) + \lambda(\hat{i} + 2\hat{j} - \hat{k})$ and $\vec{r} = (\hat{i} + \hat{j}) + \mu(-\hat{i} + \hat{j} - 2\hat{k})$ is

$ABC$ is a triangle in a plane with vertices $A(2, 3, 5)$, $B(-1, 3, 2)$ and $C(\lambda, 5, \mu)$. If the median through $A$ is equally inclined to the coordinate axes, then the value of $(\lambda^3 + \mu^3 + 5)$ is

The planes $x + y - z = 0, y + z - x = 0, z + x - y = 0$ meet :

A tetrahedron has vertices at $O(0, 0, 0)$, $A(1, 2, 1)$, $B(2, 1, 3)$ and $C(-1, 1, 2)$. Then the angle between the faces $OAB$ and $ABC$ will be

The vertices of △ABC are A(2, 0, 0), B(0, 1, 0), C(0, 0, 2). Its orthocentre is H and circumcentre is S. P is a point equidistant from A, B, C and the origin O. The y-coordinate of S is:

A symmetrical form of the line of intersection of the planes $x = ay + b$ and $z = cy + d$ is

The equation of the straight line is $\frac{x}{a} = \frac{y}{b} = \frac{z}{c}$, where the ordered triad $(a, b, c)$ is

If the distance of the point $P(43,\alpha,\beta)$, $\beta<0$, from the line $\vec{r}=4\hat{i}-\hat{k}+\mu(2\hat{i}+3\hat{k})$, $\mu\in\mathbb{R}$ along a line with direction ratios $3,-1,0$ is $13\sqrt{10}$, then $\alpha^2+\beta^2$ is equal to _____.

Perpendiculars are drawn from points on the line $\frac{x+2}{2} = \frac{y+1}{-1} = \frac{z}{3}$ to the plane $x + y + z = 3$. The feet of perpendiculars lie on the line

Consider a plane p: $\vec{r} \cdot \vec{n} = d$ (where $\vec{n}$ is not a unit vector). There are two points A($\vec{a}$) and B($\vec{b}$) lying on the same side of the plane. Reflection of A($\vec{a}$) in the plane p has the position vector:

The image of the point $(-1, 3, 4)$ in the plane $x - 2y = 0$ is

Let a line $L$ passing through the point $P(1,1,1)$ be perpendicular to the lines $\dfrac{x-4}{4}=\dfrac{y-1}{1}=\dfrac{z-1}{1}$ and $\dfrac{z-17}{1}=\dfrac{y-71}{1}=\dfrac{z}{0}$. Let the line $L$ intersect the $yz$-plane at the point $Q$. Another line parallel to $L$ and passing through the point $S(1,0,-1)$ intersects the $yz$-plane at the point $R$. Then the square of the area of the parallelogram $PQRS$ is equal to _____.

If GE and CD are mutually perpendicular, then orthocenter of △ABC must lie on:

The locus of a point which moves in such a way that its distance from the line $\frac{x}{1} = \frac{y}{1} = \frac{z}{-1}$ is twice the distance from the plane $x + y + z = 0$ is

If the distance of point of intersection of lines $\frac{x-4}{1} = \frac{y+3}{-4} = \frac{z+1}{7}$ and $\frac{x-1}{2} = \frac{y+1}{3} = \frac{z+10}{8}$ from $(1, -4, 7)$ is $a$, then $\frac{a^2}{13}$ is equal to __________.

85. Given $\dfrac{x-2}{3} = \dfrac{y+1}{2} = \dfrac{z-1}{-1}$. A point $P$ lies on the line and also on the plane $2x + 3y - z + 13 = 0$. Another line passes through $P$ and is parallel to $\dfrac{x-2}{3} = \dfrac{y+1}{2} = \dfrac{z-1}{-1}$. Find the distance (approximately) related to this configuration. (Answer: 7.483)

The equation of a line of greatest slope can be

The vertices of △ABC are A(2, 0, 0), B(0, 1, 0), C(0, 0, 2). Its orthocentre is H and circumcentre is S. P is a point equidistant from A, B, C and the origin O. PA is equal to:

If the lines $\dfrac{x-2}{1} = \dfrac{y-3}{1} = \dfrac{z-4}{-k}$ and $\dfrac{x-1}{k} = \dfrac{y-4}{2} = \dfrac{z-5}{1}$ are coplanar, then $k$ can have

The cartesian equations of the plane which passes through the point (5, 2, −4) and perpendicular to the line with direction ratios 2, 3, −1 is

The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l^2 = m^2 + n^2$ is

The coordinates of the foot of the perpendicular from the point $(1, -2, 1)$ on the plane containing the lines, $\dfrac{x+1}{6} = \dfrac{y-1}{7} = \dfrac{z-3}{8}$ and $\dfrac{x-1}{3} = \dfrac{y-2}{5} = \dfrac{z-3}{7}$, is

Ex. 61 (B): If $(\lambda, 3\lambda, \mu)$ is a point on the line $2x + y + z - 3 = 0 = x - 2y + z - 1$, then $\lambda + \mu$ is equal to

The line $\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-1}{-1}$ intersects the curve $x^2 + y^2 = r^2, z = 0$ then

Ex. 63 (A): The coordinates of a point on the line $x = 4y + 5, z = 3y - 6$ at a distance 3 from the point $(5, 3, -6)$ is/are

The square of the distance of the point ( 15 , 32 , 7) from the line x+1 = y+3 = z+5 in the direction of the vector 7 7 3 5 7 ^ ^ ^ i + 4 j + 7k is :

SD between $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is

Plane $P$ through $(5,3,0),(13,3,-2),(1,6,2)$. Distances of $A(3,4,\alpha)$ and $B(2,\alpha,a)$ from $P$ are 2 and 3. Positive value of $a$ is

If $\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$ and $\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$ intersect, then magnitude of minimum value of $8\alpha\beta$ is _____

Let the point A divide the line segment joining the points P (-1, -1, 2) and Q(5, 5, 10) internally in the ratio - -\to - -\to - -\to - -\to . If O is the origin and (OQ ⋅ OA) - , then the value of r is : 1 2 r : 1(r > 0) |OP \times OA| = 10 5

Let a = ^i + 2 ^j + k^ and b = 2 ^i + 7 ^j + 3 k^ . Let L 1 : r = (- ^i + 2 ^j + k^ ) + \lambda a , \lambda \in R and L 2 : r = ( j^ + k^ ) + \mu b , \mu \in R be two lines. If the line L 3 passes through the point of intersection of L 1 and L 2 , and is parallel to a + b , then L 3 passes through the point :

Shortest distance between the lines $\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$ and $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$ is

If the shortest distance between the line joining the points $(1, 2, 3)$ and $(2, 3, 4)$, and the line $\frac{x-1}{2} = \frac{y+1}{-1} = \frac{z-2}{0}$ is $\alpha$, then $28\alpha^2$ is equal to _____.

Foot of $\perp$ from $A(4,3,1)$ on $x-y+2z+3=0$ is $N$. $B(5,\alpha,\beta)$ on plane, area of $\triangle ABN=3\sqrt{2}$. Then $\alpha^2+\beta^2+\alpha\beta$ is equal to _____________

Let $L_1:\dfrac{x-1}{1}=\dfrac{y-2}{-1}=\dfrac{z-1}{2}$ and $L_2:\dfrac{x+1}{-1}=\dfrac{y}{2}=\dfrac{z}{1}$ be two lines. Let $L_3$ be a line passing through the point $(\alpha,\beta,\gamma)$ and be perpendicular to both $L_1$ and $L_2$. If $L_3$ intersects $L_1$, then $|5\alpha-11\beta-8\gamma|$ equals:

3D Geometry Questions (578)