Point of intersection of a line and a plane - class-XII
Description: point of intersection of a line and a plane | |
Number of Questions: 42 | |
Created by: Mohini Tyagi | |
Tags: three dimensional geometry maths applications of vector algebra three dimensional geometry - ii |
Statement-I: The point $A(3,1,6)$ is the mirror image of the point $B(1,3,4)$ in the plane $x-y+z=5$.
Statement-2: The plane $x-y+z=5$ bisects the line segment joining $A(3,1,6)$ and $B(1,3,4)$.
If the points $(1,2,3)$ and $(2,-1,0)$ lie on the opposite sides of the plane $2x+3y-2z=k$, then
If the planes $x - cy - bz = 0,cx - y + az = 0\,$ and $bx + ay - z = 0$ pass through a stright line,then the value of ${a^2} + {b^2} + {c^2} + 2abc\,$ is:
The point where the line through $A=(3, -2, 7)$ and $B= (13, 3, -8)$ meets the xy-plane
The ratio in which the plane $4x+5y-3z=8$ divides the line joining the points $(-2,1,5)$ and $(3,3,2)$ is
Let the equations of a line and a plane be $\dfrac {x+3}{2}=\dfrac {y-4}{3}=\dfrac {z+5}{2}$ and $4x-2y-z=1$, respectively, then
The ratio in which the plane $r.\left( \hat { i } -2\hat { j } +2\hat { k } \right) =17$ divides the line joining the points $-2\hat { i } +4\hat { j } +7\hat { k } $ and $3\hat { i } -5\hat { j } +8\hat { k } $ is:
Line $\vec r=\vec a+\lambda \vec b$ will not meet the plane $\vec r\cdot \vec n=q$, if-
The ratio in which the plane $\vec r\cdot (\vec i-2\vec j+3\vec k)=17$ divides the line joining the points $-2\vec i+4\vec j+7\vec k$ and $3\vec i-5\vec j+8\vec k$ is-
The plane $\vec r\cdot \vec n=q$ will contain the line $\vec r=\vec a+\lambda \vec b$, if-
The ratio in which the line segment joining the points whose position vectors are $2\hat i-4\hat j-7\hat k$ and $-3\hat i+5\hat j-8\hat k$ is divided by the plane whose equation is $\hat r\cdot (\hat i-2\hat j+3\hat k)=13$ is-
Which of the following lines lie on the plane $x+2y-z=0$?
Find the ratio in which the segment joining $(1, 2, -1)$ and $(4, -5, 2)$ is divided by the plane $2x - 3y + z = 4$
If the given planes $ax+by+cz+d=0$ and $ax+by+cz+d=0$ be mutually perpendicular, then
The ratio in which the joint of $(2, 1, 5), (3, 4, 3)$ is divided by the plane $2x + 2y - 2z - 1 = 0$
A straight line $\overline { r } =\overline { a } +\lambda \overline { b } $ meets the plane $\overline { r } .\overline { n } =0$ at a point $p$. The position vector of $p$ is
The distance of the point $(-1,-5,-10)$ from the point of intersection of the line $\dfrac{x-2}{2}=\dfrac{y+1}{4}=\dfrac{z-2}{12}$ and the plane $x-y+z=5$ is
The point of intersection of the line joining the points $(2,0,2)$ and $(3,-1,3)$ and the plane $x-y+z=1$ is
The expression in the vector form for the point $\vec { r } _ { 1 }$ of intersection of the plane $\vec { r } \cdot \vec { n } = d$ and the perpendicular line $\vec { r } = \vec { r } _ { 0 } + \hat { n }$ where $t$ is a parameter given by -
If the line $\displaystyle \frac{x - 1}{1} = \frac{y + 1}{-2} = \frac{z + 1}{\lambda}$ lies in the plane $\displaystyle 3x - 2y + 5z = 0$ then $\displaystyle \lambda$ is
The Foot of the $\displaystyle \perp$ from origin to the plane $\displaystyle 3x + 4y - 6z + 1 = 0$ is
The co-ordinate of a point where the line $(2, -3, 1)$ and $(3, -4, -5)$ cuts the plane $2x + y + z = 7$ are $(1, k, 7)$ then value of $k$ equals
The condition that the line $\displaystyle \frac{x-{\alpha }'}{l}=\frac{y -{\beta }'}{m}=\frac{z-{\gamma }'}{n}$ in the plane $Ax + By + Cz + D = 0$ is
Consider a point $P (1, 2, 3)$, plane $ \pi : x + y + z = 11 $ and the line $ L : \displaystyle \frac{x+1}{1}=\displaystyle \frac{y-12}{-2} = \displaystyle \frac{z-7}{2} $ The foot of the $ \perp $ drawn from the point P meet the plane $ \pi $ at M, then co-ordinate of M is
The point of intersection of the line $\dfrac{x-1}{3}=\dfrac{y+2}{4}=\dfrac{z-3}{-2}$ and the plane $2x-y+3z-1=0$, is
Find the point where the line of intersection of the planes $x-2y+z=1$ and $x+2y-2z=5$ intersects the plane $3x+2y+z+6=0$.
The line joining the points $\left (2, -3, 1 \right )$ and $\left (3, -4, -5 \right )$ cuts a coordinate plane at the point.
Let line L: $\displaystyle \frac{x-1}{2} = \frac{y - 1}{1} = \frac{z - 0}{4} $ & Plane P: $x + 2y - z = 3$
Then which of the following is true?
If a line which passes through the point $A(0,\,1,\,2)$ and makes angle $\displaystyle\frac{\pi}{4},\,\displaystyle\frac{\pi}{4},\,\displaystyle\frac{\pi}{2}$ with $x,\,y,\,&\,z$ axes respectively. The line meets the plane $x+y+z=0$ at point $B$. The length $\sqrt{2}AB$ is equal to
The ratio in which the plane $\vec{r}.(\hat{i}-2\hat{j}+3\hat{k})=17$ divides the line joining the points $(-2\hat{i}+4\hat{j}+7\hat{k})$ and $(3\hat{i}-5\hat{j}+8\hat{k})$ is
A straight line $\overrightarrow { r } =\overrightarrow { a } +\lambda \overrightarrow { b } $ meets the plane $\overrightarrow { r } .\overrightarrow { n } =0$ in $P$. The position vector of $P$ is
The value of $k$ such that $\displaystyle \dfrac{{x}-4}{1}=\dfrac{{y}-2}{1}=\dfrac{{z}-{k}}{2}$ lies in the plane $2x-4y+{z}=7$ is
The plane $x-2y+z-6=0$ and the line $\displaystyle\frac{x}{1}=\displaystyle\frac{y}{2}=\displaystyle\frac{z}{3}$ are related as.
The plane ax + by + cz = 1 meets the coordinate axes in A, B and C. The centroid of $\triangle ABC$ is
The plane $\frac{x}{y}+\frac{y}{3}+\frac{z}{4}$ =1 cutes the axes in A,B,C, then the are of the $\Delta ABC$ is;
Perpendicular is drawn from the point $(0,3,4)$ to the plane $2x -2y + z + (-10) = 0$, then co-ordinates of the foot of the L's are
Let the line $\displaystyle \frac{x-2}{3}= \frac{y-1}{-5}= \frac{z+2}{2}$ lie in the plane $x+3y-\alpha z+\beta = 0$. Then $\left ( \alpha ,\beta \right )$ equals :
The line $x -2y + 4z + 4 = 0$, $x + y + z - 8 = 0$ intersects the plane $x - y + 2z + 1 = 0$ at the point
$L: \displaystyle \frac{x\, +\, 1}{2}= \frac{y\, +\, 1}{3}= \frac{z\, +\, 1}{4}$
$\pi _{1}:\, x\, +\, 2y\, +\, 3z= 14,\, \pi _{2}:\, 2x\, -\, y\, +\, 3z= 27$
A line with positive direction cosines passes through the point $\displaystyle P\left ( 2,-1,2 \right )$ and makes equal angles with the coordinates axis. The line meet the plane $\displaystyle 2x+y+z=9$ at ponit $Q$.
The length of the line segment $PQ$ equals.
Find the point where the line of intersection of the planes $ x - 2y + z = 1$ and $x + 2y - 2z = 5$, intersects the plane $2x + 2y + z + 6 = 0$
The line passing through the points $(5, 1, a)$ and $(3, b, 1)$ crosses the $yz$-plane at the point $\left (0,\dfrac{17}{2},\dfrac{-13}{2}\right)$. Then,