Intersection of a Line and a Plane

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JEE Main 2016

If the line, $\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}$ lies in the plane, $l x + m y - z = 9$ , then $l^{2} + m^{2}$ is equal to :

5

(B)

2

(C)

26

(D)

18

JEE Main 2016

LEVELJEE Main

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

(A)

\frac{10}{3}

(B)

\frac{20}{3}

(C)

310

(D)

103

JEE Main 2015

LEVELJEE Main

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

(A)

321

(B)

13

(C)

214

(D)

8

JEE Main 2014

LEVELJEE Main

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

(A)

\frac{x - 3}{3} = \frac{y + 5}{1} = \frac{z - 2}{- 5}

(B)

\frac{x - 3}{- 3} = \frac{y + 5}{- 1} = \frac{z - 2}{5}

(C)

\frac{x + 3}{3} = \frac{y - 5}{1} = \frac{z - 2}{- 5}

(D)

\frac{x + 3}{- 3} = \frac{y - 5}{- 1} = \frac{z + 2}{5}

JEE Main 2011

LEVELJEE Main

If the angle between the line $x = \frac{y - 1}{2} = \frac{z - 3}{λ}$ and the plane $x + 2 y + 3 z = 4$ is $cos^{- 1} \frac{5}{14}$ , then $λ$ equals

(A)

\frac{3}{2}

(B)

\frac{2}{5}

(C)

\frac{5}{3}

(D)

\frac{2}{3}

JEE Main 2010

LEVELJEE Main

Statement-1 : 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 $B (1, 3, 4)$ .

(A)

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(B)

Statement-1 is true, Statement-2 is false.

(C)

Statement-1 is false, Statement-2 is true.

(D)

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

JEE Main 2009

LEVELJEE Main

Let the line $\frac{x - 2}{3} = \frac{y - 1}{- 5} = \frac{z + 2}{2}$ lie in the plane $x + 3 y - α z + β = 0$ . Then $(α, β)$ equals

(A)

(- 6, 7)

(B)

(5, - 15)

(C)

(- 5, 5)

(D)

(6, - 17)

JEE Main 2005

LEVELJEE Main

If the angle $θ$ between the line $\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$ and the plane $2 x - y + λ z + 4 = 0$ is such that $sin θ = \frac{1}{3}$ , then the value of $λ$ is

(A)

5/3

(B)

- 3/5

(C)

3/4

(D)

- 4/3

JEE Advanced 2012

LEVELJEE Advanced

The point $P$ is the intersection of the straight line joining the points $Q (2, 3, 5)$ and $R (1, - 1, 4)$ with the plane $5 x - 4 y - z = 1$ . If $S$ is the foot of the perpendicular drawn from the point $T (2, 1, 4)$ to $QR$ , then the length of the line segment $P S$ is

(A)

\frac{1}{2}

(B)

2

(C)

(D)

22

JEE Advanced 2012

LEVELJEE Main

The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac{2}{3}$ from the point $(3, 1, - 1)$ is

(A)

5 x - 11 y + z = 17

(B)

2 x + y = 32 - 1

(C)

x + y + z = 3

(D)

x - 2 y = 1 - 2

JEE Advanced 2010

LEVELJEE Main

If the distance of the point $P (1, - 2, 1)$ from the plane $x + 2 y - 2 z = α$ , where $α > 0$ , is 5, then the foot of the perpendicular from $P$ to the plane is

(A)

(\frac{8}{3}, \frac{4}{3}, - \frac{7}{3})

(B)

(\frac{4}{3}, \frac{4}{3}, \frac{1}{3})

(C)

(\frac{1}{3}, \frac{2}{3}, \frac{10}{3})

(D)

(\frac{2}{3}, \frac{1}{3}, \frac{5}{2})

JEE Advanced 2009

LEVELJEE Main

Let $P (3, 2, 6)$ be a point in space and $Q$ be a point on the line $r = (\hat{i} - \hat{j} + 2 \hat{k}) + μ (- 3 \hat{i} + \hat{j} + 5 \hat{k})$ . Then the value of $μ$ for which the vector $P Q$ is parallel to the plane $x - 4 y + 3 z = 1$ is

(A)

\frac{1}{4}

(B)

- \frac{1}{4}

(C)

\frac{1}{8}

(D)

- \frac{1}{8}

JEE Advanced 2009

LEVELJEE Main

A line with positive direction cosines passes through the point $P (2, - 1, 2)$ and makes equal angles with the coordinate axes. The line meets the plane $2 x + y + z = 9$ at point $Q$ . The length of the line segment $P Q$ equals

(A)

(B)

2

(C)

3

(D)

JEE Advanced 2003S

LEVELJEE Main

The value of $k$ such that $\frac{x - 4}{1} = \frac{y - 2}{1} = \frac{z - k}{2}$ lies in the plane $2 x - 4 y + z = 7$ , is

(A)

7

(B)

- 7

(C)

no real value

(D)

4

JEE Main 2017

LEVELJEE Advanced

If the image of the point $P (1, - 2, 3)$ in the plane, $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line, $\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$ is Q, then PQ is equal to:

(A)

35

(B)

242

(C)

42

(D)

65

JEE Main 2018 (Paper 1)

LEVELJEE Main

The length of the projection of the line segment joining the points $(5, - 1, 4)$ and $(4, - 1, 3)$ on the plane, $x + y + z = 7$ is :

(A)

2/3

JEE Main 2018 (15 April Shift 1)

LEVELJEE Main

An angle between the plane, $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$ , is

(A)

sin^{- 1} (3/ 17)

(B)

cos^{- 1} (3/17)

(C)

sin^{- 1} (3/17)

(D)

cos^{- 1} (3/ 17)

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

Let A be a point on the line $r = (1 - 3 μ) \hat{i} + (μ - 1) \hat{j} + (2 + 5 μ) \hat{k}$ and B(3, 2, 6) be a point in the space. Then the value of $μ$ for which the vector $A B$ is parallel to the plane $x - 4 y + 3 z = 1$ is :

(A)

\frac{1}{2}

(B)

-\frac{1}{4}

(C)

\frac{1}{4}

(D)

\frac{1}{8}

JEE Main 2019 (12 January)

LEVELJEE Main

If an angle between the line, $\frac{x + 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{- 2}$ and the plane, $x - 2 y - k z = 3$ is $cos^{- 1} (\frac{2 2}{3})$ , then a value of $k$ is:

(A)

- \frac{5}{3}

(B)

\frac{3}{5}

(C)

\frac{5}{3}

(D)

- \frac{3}{5}

JEE Main 2019 (10 January)

LEVELJEE Main

On which of the following lines lies the point of intersection of the line, $\frac{x - 4}{2} = \frac{y - 5}{2} = \frac{z - 3}{1}$ and the plane, $x + y + z = 2$ ?

(A)

\frac{x - 2}{2} = \frac{y - 3}{2} = \frac{z + 3}{3}

(B)

\frac{x - 4}{1} = \frac{y - 5}{1} = \frac{z - 5}{- 1}

(C)

\frac{x - 1}{1} = \frac{y - 3}{2} = \frac{z + 4}{- 5}

(D)

\frac{x + 3}{3} = \frac{4 - y}{3} = \frac{z + 1}{- 2}

JEE Main 2019 (09 April Shift 1)

LEVELJEE Main

If the line, $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4}$ meets the plane, $x + 2 y + 3 z = 15$ at a point $P$ , then the distance of $P$ from the origin is

(A)

9/2

(B)

25

(C)

5 /2

(D)

7/2

JEE Main 2019 (10 April Shift 2)

LEVELJEE Advanced

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

JEE Main 2019 (12 April Shift 1)

LEVELJEE Main

If the line $\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z - 1}{- 1}$ intersects the plane $2 x + 3 y - z + 13 = 0$ at a point P and the plane $3 x + y + 4 z = 16$ at a point Q, then PQ is equal to :

(A)

214

(B)

14

(C)

27

(D)

14

JEE Main 2020 - 4 Sep (Evening)

LEVELJEE Main

The distance of the point $(1, - 2, 3)$ from the plane $x - y + z = 5$ measured parallel to the line $\frac{x}{2} = \frac{y}{3} = \frac{z}{- 6}$ is:

(A)

7

(B)

7/5

(C)

1

(D)

1/7

JEE Main 2021 (31 Aug Shift 2)

LEVELJEE Main

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line, 2x−3​=−1y+2​=3z+4​ lies in the plane, lx+my−z=9, then l2+m2 is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance of the point (1,−5,9) from the plane x−y+z=5 measured along the line x=y=z is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance of the point (1,0,2) from the point of intersection of the line 3x−2​=4y+1​=12z−2​ and the plane x−y+z=16, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The image of the line 3x−1​=1y−3​=−5z−4​ in the plane 2x−y+z+3=0 is the line:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the angle between the line x=2y−1​=λz−3​ and the plane x+2y+3z=4 is cos−1145​​, then λ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the line 3x−2​=−5y−1​=2z+2​ lie in the plane x+3y−αz+β=0. Then (α,β) equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the angle θ between the line 1x+1​=2y−1​=2z−2​ and the plane 2x−y+λ​z+4=0 is such that sinθ=31​, then the value of λ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of a plane passing through the line of intersection of the planes x+2y+3z=2 and x−y+z=3 and at a distance 3​2​ from the point (3,1,−1) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the distance of the point P(1,−2,1) from the plane x+2y−2z=α, where α>0, is 5, then the foot of the perpendicular from P to the plane is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of k such that 1x−4​=1y−2​=2z−k​ lies in the plane 2x−4y+z=7, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the image of the point P(1,−2,3) in the plane, 2x+3y−4z+22=0 measured parallel to the line, 1x​=4y​=5z​ is Q, then PQ is equal to:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The length of the projection of the line segment joining the points (5,−1,4) and (4,−1,3) on the plane, x+y+z=7 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An angle between the plane, x+y+z=5 and the line of intersection of the planes, 3x+4y+z−1=0 and 5x+8y+2z+14=0, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If an angle between the line, 2x+1​=1y−2​=−2z−3​ and the plane, x−2y−kz=3 is cos−1(322​​), then a value of k is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } On which of the following lines lies the point of intersection of the line, 2x−4​=2y−5​=1z−3​ and the plane, x+y+z=2 ?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line, 2x−1​=3y+1​=4z−2​ meets the plane, x+2y+3z=15 at a point P, then the distance of P from the origin is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line 3x−2​=2y−1​=−1z−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 :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The distance of the point (1,−2,3) from the plane x−y+z=5 measured parallel to the line 2x​=3y​=−6z​ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Suppose the line αx−2​=−5y−2​=2z+2​ lies on the plane x+3y−2z+β=0. Then (α+β) is equal to .

If the line, $\frac{x - 3}{2} = \frac{y + 2}{- 1} = \frac{z + 4}{3}$ lies in the plane, $l x + m y - z = 9$ , then $l^{2} + m^{2}$ is equal to :

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

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

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

If the angle between the line $x = \frac{y - 1}{2} = \frac{z - 3}{λ}$ and the plane $x + 2 y + 3 z = 4$ is $cos^{- 1} \frac{5}{14}$ , then $λ$ equals

Let the line $\frac{x - 2}{3} = \frac{y - 1}{- 5} = \frac{z + 2}{2}$ lie in the plane $x + 3 y - α z + β = 0$ . Then $(α, β)$ equals

If the angle $θ$ between the line $\frac{x + 1}{1} = \frac{y - 1}{2} = \frac{z - 2}{2}$ and the plane $2 x - y + λ z + 4 = 0$ is such that $sin θ = \frac{1}{3}$ , then the value of $λ$ is

The equation of a plane passing through the line of intersection of the planes $x + 2 y + 3 z = 2$ and $x - y + z = 3$ and at a distance $\frac{2}{3}$ from the point $(3, 1, - 1)$ is

If the distance of the point $P (1, - 2, 1)$ from the plane $x + 2 y - 2 z = α$ , where $α > 0$ , is 5, then the foot of the perpendicular from $P$ to the plane is

The value of $k$ such that $\frac{x - 4}{1} = \frac{y - 2}{1} = \frac{z - k}{2}$ lies in the plane $2 x - 4 y + z = 7$ , is

If the image of the point $P (1, - 2, 3)$ in the plane, $2 x + 3 y - 4 z + 22 = 0$ measured parallel to the line, $\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$ is Q, then PQ is equal to:

The length of the projection of the line segment joining the points $(5, - 1, 4)$ and $(4, - 1, 3)$ on the plane, $x + y + z = 7$ is :

An angle between the plane, $x + y + z = 5$ and the line of intersection of the planes, $3 x + 4 y + z - 1 = 0$ and $5 x + 8 y + 2 z + 14 = 0$ , is

If an angle between the line, $\frac{x + 1}{2} = \frac{y - 2}{1} = \frac{z - 3}{- 2}$ and the plane, $x - 2 y - k z = 3$ is $cos^{- 1} (\frac{2 2}{3})$ , then a value of $k$ is:

On which of the following lines lies the point of intersection of the line, $\frac{x - 4}{2} = \frac{y - 5}{2} = \frac{z - 3}{1}$ and the plane, $x + y + z = 2$ ?

If the line, $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 2}{4}$ meets the plane, $x + 2 y + 3 z = 15$ at a point $P$ , then the distance of $P$ from the origin is

If the line $\frac{x - 2}{3} = \frac{y - 1}{2} = \frac{z - 1}{- 1}$ intersects the plane $2 x + 3 y - z + 13 = 0$ at a point P and the plane $3 x + y + 4 z = 16$ at a point Q, then PQ is equal to :

The distance of the point $(1, - 2, 3)$ from the plane $x - y + z = 5$ measured parallel to the line $\frac{x}{2} = \frac{y}{3} = \frac{z}{- 6}$ is:

Suppose the line $\frac{x - 2}{α} = \frac{y - 2}{- 5} = \frac{z + 2}{2}$ lies on the plane $x + 3 y - 2 z + β = 0$ . Then $(α + β)$ is equal to .