3D Geometry Questions (578)

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:

Distance of $(-1,2,3)$ from $\vec{r}\cdot(\hat{i}-2\hat{j}+3\hat{k})=10$ measured $\parallel$ to SD line between $\vec{r}=(\hat{i}-\hat{j})+\lambda(2\hat{i}+\hat{k})$ and $\vec{r}=(2\hat{i}-\hat{j})+\mu(\hat{i}-\hat{j}+\hat{k})$ is

Let the equation of the plane P containing the line $x + 10 = \frac{8-y}{2} = z$ be $ax + by + 3z = 2(a+b)$ and the distance of the plane P from the point $(1, 27, 7)$ be $c$. Then $a^2 + b^2 + c^2$ is equal to _____.

Image of $P(2,3,5)$ in $2x+y-3z=6$ is $(\alpha,\beta,\gamma)$. Then $\alpha+\beta+\gamma$ is equal to

A plane passes through $(1, -2, 1)$ and is perpendicular to two planes $2x - 2y + z = 0$ and $x - y + 2z = 4$. The distance of the plane from the origin $(0, 2)$ is

Let $\theta$ be the angle between the planes $P_1 = \vec{r}\cdot(\hat{i}+\hat{j}+2\hat{k})=9$ and $P_2 = \vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=15$. Let L be the line that meets $P_2$ at the point $(4, -2, 5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between L and $P_2$ then $(\tan^2\theta)(\cot^2\alpha)$ is equal to _____.

Two vertices of $\triangle ABC$: $(2,4,6)$, $(0,-2,-5)$; centroid $(2,1,-1)$. Third vertex $C=(4,1,2)$. Image of $C$ in $x+2y+4z=11$ is $(\alpha,\beta,\gamma)$. $\alpha\beta+\beta\gamma+\gamma\alpha$ is equal to

Let $A(x,y,z)$ be a point in $xy$-plane, which is equidistant from three points $P(0,3,2)$, $Q(2,0,3)$ and $R(0,0,1)$. Let $B=(1,4,-1)$ and $C=(2,0,-2)$. Then among the statements (S1): $\triangle ABC$ is an isosceles right angled triangle, and (S2): the area of $\triangle ABC$ is $\dfrac{9\sqrt{2}}{2}$

Let P be the foot of the perpendicular from the point Q(10, -3, -1) on the line x-3 y-2 z+1 7 = -1 = -2 . Then the area of the right angled triangle P QR, where R is the point (3, -2, 1), 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 $14l^2$ is equal to

If the equation of the plane passing through the point $(1,1,2)$ and perpendicular to the line $x - 3y + 2z - 1 = 0$, $4x - y + z = 0$ is $Ax + By + Cz = 1$, then $140(C - B + A)$ is equal to _____.

Let a straight line L pass through the point P (2, -1, 3) and be perpendicular to the lines y+1 x-1 2 = 1 = z-3 -2 and x-3 y-2 z+2 1 = 3 = 4 . If the line L intersects the yz-plane at the point Q, then the distance between the points P and Q is :

Let the co-ordinates of one vertex of $\triangle ABC$ be $A(0, 2, \alpha)$ and the other two vertices lie on the line $\frac{x+\alpha}{5} = \frac{y-1}{2} = \frac{z+4}{3}$. For $\alpha \in \mathbb{Z}$, if the area of $\triangle ABC$ is 21 sq. units and the line segment BC has length $2\sqrt{21}$ units, then $\alpha^2$ is equal to _____.

Let L : y-1 y z+4 1 x-1 3 = -1 = z+1 0 and L : 2 x-2 2 = 0 = \alpha ,\alpha \in R , be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A(1, 1, -1) on L , then the value of 26\alpha( PB) is _________ 2 2

Image of $P(1,2,3)$ in plane $2x-y+z=9$ is $Q$. $R=(6,10,7)$. Then square of area of $\triangle PQR$ is _______

Ex. 61 (A): If the line \(\frac{x-1}{1} = \frac{y+1}{-2} = \frac{z+1}{\lambda}\) lies in the plane \(3x - 2y + 5z = 0\), then \(\lambda\) is equal to

$P$ = intersection of $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ with $x+y+z=2$. Distance from $P$ to $3x-4y+12z=32$ is $q$. Then $q$ and $2q$ are roots of

Let $L_1:\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and $L_2:\dfrac{x-2}{3}=\dfrac{y-4}{4}=\dfrac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$?

The equation of a plane is \(x + 2y + 2z + 7 = 0\) and the equation of a sphere is \(x^2 + y^2 + z^2 + 2x - 2y - 4z - 19 = 0\). The radius of the circle of intersection of the plane and sphere is:

The shortest distance between the lines $\frac{x-5}{1} = \frac{y-2}{2} = \frac{z-4}{-3}$ and $\frac{x+3}{1} = \frac{y+5}{4} = \frac{z-1}{-5}$ is

If an angle between the line, \(\dfrac{x+1}{2} = \dfrac{y-2}{1} = \dfrac{z-3}{-2}\) and the plane, \(x - 2y - kz = 3\) is \(\cos^{-1}(2\sqrt{2}/3)\), then a value of \(k\) is ______ (up to three decimal places).

Let a unit vector $\overrightarrow{OP}$ make angles $\alpha, \beta, \gamma$ with the positive directions of the co-ordinate axes OX, OY, OZ respectively, where $\beta \in \left(0, \frac{\pi}{2}\right)$. $\overrightarrow{OP}$ is perpendicular to the plane through points $(1,2,3)$, $(2,3,4)$ and $(1,5,7)$. Then which one of the following is true?

Find the length of the projection of vector $\vec{AB}$ onto the normal vector to a plane, where $A = (1, 2, -1)$, $B = (3, 5, 5)$, and the normal vector is $3\vec{i} - 4\vec{j} + 12\vec{k}$.

Line $L$ through $(0,1,2)$ intersects $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and is parallel to $2x+y-3z=4$. Distance of $P(1,-9,2)$ from $L$ is

Image of $P(1,2,6)$ in plane through $A(1,2,0)$, $B(1,4,1)$, $C(0,5,1)$ is $Q(\alpha,\beta,\gamma)$. Then $\alpha^2+\beta^2+\gamma^2$ is equal to

Let the plane $P: 8x + \alpha_1 y + \alpha_2 z + 12 = 0$ be parallel to the line $L: \frac{x+2}{2} = \frac{y-3}{3} = \frac{z+4}{5}$. If the intercept of P on the y-axis is 1, then the distance between P and L is:

Let the plane P pass through the intersection of the planes $2x + 3y - z = 2$ and $x + 2y + 3z = 6$, and be perpendicular to the plane $2x + y - z + 1 = 0$. If $d$ is the distance of P from the point $(-7, 1, 1)$, then $d^2$ is equal to:

Let a line parallel to \(z\)-axis passing through a point \(P(3, 4, a)\) intersects the plane \(x - 2y + 2z = a^2 + 4a + 1\) at \(Q\) where \(a \in R\). If least area of \(\triangle OPQ\) is equal to \(\left(\dfrac{p}{q}\right)\) where \(p\) and \(q\) are co-prime numbers, then find the value of \((p + q)\).

Plane containing $x+2y+3z-4=0=2x+y-z+5$ and $\perp$ to $\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k})$ is $ax+by+cz=4$. Then $a-b+c$ is equal to