HARD

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Let $P_{1}$ and $P_{2}$ be two planes given by

$P_{1} : 10 x + 15 y + 12 z - 60 = 0,$

$P_{2} : - 2 x + 5 y + 4 z - 20 = 0$ .

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?

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Important Questions on Three Dimensional Geometry

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

If the angle between the line

2 (x + 1) = y = z + 4

and the plane

2 x - y + \sqrt{λ} z + 4 = 0

\frac{π}{6}

, then the value of

λ

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

The distance of the point

(1, - 5, 9)

from the plane

x - y + z = 5

measured along the line

x = y = z

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

Acute angle between the line

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

and the plane

3 x - 4 y - z + 5 = 0

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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 ?

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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

P Q

is equal to

EASY

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

If the line joining the points

(- 1, 2, 5)

and

(3, 4, - 10)

intersects the

x y

-plane at the point,

(x, y, z),

then

\frac{y}{x}

is equal to______.

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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

\cos^{- 1} (\frac{2 \sqrt{2}}{3})

, then a value of

k

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

A plane containing the point

(3, 2, 0)

and the line

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

also contains the point

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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 :

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

Let

P

be the image of the point

(3, 1, 7)

with respect to the plane

x - y + z = 3 .

Then the equation of the plane passing through

P

and containing the straight line

\frac{x}{1} = \frac{y}{2} = \frac{z}{1}

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

Let

L_{1}

and

L_{2}

be the following straight lines.

L_{1} : \frac{x - 1}{1} = \frac{y}{- 1} = \frac{z - 1}{3}

and

L_{2} : \frac{x - 1}{- 3} = \frac{y}{- 1} = \frac{z - 1}{1}

Suppose the straight line

L : \frac{x - α}{l} = \frac{y - 1}{m} = \frac{z - γ}{- 2}

lies in the plane containing

L_{1}

and

L_{2}

, and passes through the point of intersection of

L_{1}

and

L_{2}

. If the line

L

bisects the acute angle between the lines

L_{1}

and

L_{2}

, then which of the following statements is/are TRUE?

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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,

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

θ

is the angle between the line

\vec{r} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (\hat{i} - \hat{j} + 2 \hat{k}), λ \in R

and the plane

\vec{r} \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 4,

then

a

value of

\cos θ

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

O \equiv (0, 0, 0), P \equiv (1, \sqrt{2}, 1),

then the acute angles made by the line

O P

with

X O Y

Y O Z, Z O X

planes are, respectively

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

A line

L

is passing through the point

A

whose position vector is

\hat{i} + 2 \hat{j} - 3 \hat{k}

and parallel to the vector

2 \hat{i} + \hat{j} + 2 \hat{k}

. A plane

π

is passing through the points

\hat{i} + \hat{j} + \hat{k}, \hat{i} - \hat{j} - \hat{k}

and parallel to the vector

\hat{i} - 2 \hat{j}

. Then the point where this plane

π

meets the line

L

HARD

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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}

Q

, then

P Q

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

Let

A

be a point on the line

\vec{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

\vec{A B}

is parallel to the plane

x - 4 y + 3 z = 1

MEDIUM

Mathematics>Coordinate Geometry>Three Dimensional Geometry>Interaction of a Line and a Plane

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

Let P1 and P2 be two planes given by P1:10x+15y+12z-60=0, P2:-2x+5y+4z-20=0. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1 and P2?

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Let $P_{1}$ and $P_{2}$ be two planes given by

$P_{1} : 10 x + 15 y + 12 z - 60 = 0,$

$P_{2} : - 2 x + 5 y + 4 z - 20 = 0$ .

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?