3D Geometry Questions - JEE

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

The equation of a plane is $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$. The plane passes through the point $(3, 2, 1)$ and meets the axes at $A(a, 0, 0)$, $B(0, b, 0)$ and $C(0, 0, c)$. The locus of the point of intersection of planes through $A$, $B$ and $C$ parallel to the $yz$-, $zx$- and $xy$-planes respectively is:

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $(-\lambda^2, 1, 1)$, $(1, -\lambda^2, 1)$ and $(1, 1, -\lambda^2)$ also passes through the point $(-1, -1, 1)$. Then $S$ is equal to:

A line segment has direction cosines $l, m, n$. If the line makes an angle of $45°$ with the $x$-axis and $120°$ with the $y$-axis, then the angle made by the line with the positive $z$-axis is:

Let the lines $L_1:\vec{r}=\hat{i}+2\hat{j}+3\hat{k}+\lambda(2\hat{i}+3\hat{j}+4\hat{k})$ and $L_2:\vec{r}=(4\hat{i}+\hat{j})+\mu(5\hat{i}+2\hat{j}+\hat{k})$ intersect at the point $R$. Let $P$ and $Q$ be points on $L_1$ and $L_2$ respectively such that $|\overrightarrow{PR}|=\sqrt{29}$ and $|\overrightarrow{PQ}|=\sqrt{47/3}$. If $P$ lies in the first octant, then $27(QR)^2$ is equal to

The distance of the point $(1, -2, 4)$ from the plane passing through the point $(1, 2, 2)$ and perpendicular to the planes $x - y + 2z = 3$ and $2x - 2y + z + 12 = 0$ is

If $10y - 8x - (x^2 + y^2 + z^2) = 40, P_1 = \max\left\{\sqrt{(x+2)^2 + (y-3)^2 + z^2}\right\}, P_2 = \min\left\{\sqrt{(x+2)^2 + (y-3)^2 + z^2}\right\}$, then $P_1 - P_2$ is __________.

The perpendicular distance from the origin to the plane containing the two lines, $\dfrac{x+2}{3} = \dfrac{y-2}{5} = \dfrac{z+5}{7}$ and $\dfrac{x-1}{1} = \dfrac{y-4}{4} = \dfrac{z+4}{7}$, is ______ (up to three decimal places).

If the direction cosines of two lines are such that l + m + n = 0 and l^2 + m^2 - n^2 = 0, then the angle between them is

If the length of the perpendicular from the point $(\beta, 0, \beta)$ ($\beta \neq 0$) to the line, $\dfrac{x}{1} = \dfrac{y-1}{0} = \dfrac{z+1}{-1}$ is $\sqrt{\dfrac{3}{2}}$, then $|\beta|$ is equal to ______.

The vector equation of plane which is at a distance of 8 units from the origin and which is normal to the vector $2\vec{i} + \vec{j} + 2\vec{k}$ is $\vec{r} \times (2\vec{i} + \vec{j} + 2\vec{k}) = l$, where $l$ is equal to

The equation of a line of greatest slope on an inclined plane can be represented as:

If $\cos^2\theta + \cos^2\theta + \cos^2\alpha = 1$ (sum of squares of direction cosines = 1), find the range of $\theta$ given that $0 \leq 2\cos^2\theta \leq \frac{1}{2}$.

The shortest distance between the lines x + 1, y + 1, z + 1 is

If the point $(2, \alpha, \beta)$ lies on the plane which passes through the points $(3, 4, 2)$ and $(7, 0, 6)$ and is perpendicular to the plane $2x - 5y = 15$, then $2\alpha - 3\beta$ is equal to ______.

The equation of the plane containing the planes $2x - 5y + z = 3$ and $x + y + 4z = 5$ and parallel to the plane $x + 3y + 6z = 1$ is

Equation of plane passing through the points C_1(2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z - 1 = 0, is

If direction ratios of a line are $l, m, n$ and the direction ratios of normal to a plane are $a, b, c$, then the condition for the line to be parallel to the plane 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 $\vec{r}=(-3\hat{i}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}+5\hat{k})$ and $\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu(-\hat{i}+3\hat{j}+2\hat{k})$ is

Equation of the line of the shortest distance between the lines $\dfrac{x}{1} = \dfrac{y}{-1} = \dfrac{z}{1}$ and $\dfrac{x-1}{0} = \dfrac{y+1}{-2} = \dfrac{z}{1}$ is

The plane containing the line $\dfrac{x-1}{1} = \dfrac{y-2}{2} = \dfrac{z-3}{3}$ and parallel to the line $\dfrac{x}{1} = \dfrac{y}{1} = \dfrac{z}{1}$ passes through the point

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 $L_2$ is 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

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:

3D Geometry Questions (578)