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Byju's Answer

Standard XII

Mathematics

Angle between Two Line Segments

Consider the ...

Question

Consider the line $L$ given by the equation $\frac{x - 3}{2} = \frac{y - 1}{1} = \frac{z - 2}{1} .$ Let $Q$ be the mirror image of the point $(2, 3, - 1)$ with respect to $L .$ Let a plane $P$ be such that it passes through $Q,$ and the line $L$ is perpendicular to $P .$ Then which of the following points is on the plane $P$ :

(- 1, 1, 2)

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(1, 2, 2)

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(1, 1, 1)

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(1, 1, 2)

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Solution

The correct option is B $(1, 2, 2)$
$L$ is normal to $P$ and plane $P$ will pass through $(2, 3, - 1)$
Equation of $P$ is $2 x + y + z =$ $λ$ $= 6$ which is satisfied by $(1, 2, 2) .$

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Similar questions

Q. In

R^{3},

let

P_{1} : x - y + z = 3

and

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

be the equation of plane and straight line respectively. Let

Q

be the image of the point

(- 1, 2, 3)

with respect to the plane

P_{1} .

Then the equation of the plane

P_{2}

passing through

Q

and containing the line

L

Q. In

R^{3},

let

P_{1} : x - y + z = 3

and

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

be the equation of plane and straight line respectively. Let

Q

be the image of the point

(- 1, 2, 3)

with respect to the plane

P_{1} .

Then the equation of the plane

P_{2}

passing through

Q

and containing the line

L

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

Q. Let

P_{1} : x + y + z = 0

P_{2} : x + 2 y + 3 z = 0

P_{3} : x + 3 y + 5 z = 0

be three planes

L_{1}

is the line of intersection of

P_{1}

P_{2} . Let P (2, 1, - 1)

be point on

P_{3}

. If

(α, β, γ)

is image of point

P

in the line

L_{1}

then the value of

| α + β + γ |

Q. Assertion :

Consider three planes:

P_{1} : x - y + z = 1 P_{2} : x + y - z = 1 P_{3} : x - 3 y + 3 z = 2

Let

L_{1}, L_{2}, L_{3}

be the lines of intersection of the planes

P_{2}

and

P_{3}, P_{3}

and

P_{1}, P_{1}

and

P_{2},

respectively.

At least two of the lines

L_{1}, L_{2}

and

L_{3}

are non-parallel. Reason: The three planes do not have a common point.

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Consider the line L given by the equation x−32=y−11=z−21. Let Q be the mirror image of the point (2,3,–1) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P:

The correct option is B (1,2,2)L is normal to P and plane P will pass through (2,3,–1) Equation of P is 2x+y+z= λ =6 which is satisfied by (1,2,2).

The correct option is B $(1, 2, 2)$
$L$ is normal to $P$ and plane $P$ will pass through $(2, 3, - 1)$
Equation of $P$ is $2 x + y + z =$ $λ$ $= 6$ which is satisfied by $(1, 2, 2) .$