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Byju's Answer

Standard XII

Mathematics

Equation of a Plane Passing through a Point and Parallel to the Two Given Vectors

Let a plane P...

Question

Let a plane $P_{1}$ containing the line $L_{1} : \frac{x - 1}{1} = \frac{y - 1}{- 1} = \frac{z - 1}{- 1}$ and $P_{2}$ is another plane containing the line $L_{1}$ and passes through point $(0, 1, 1),$ if direction ratio of normal to the plane $P_{1}$ is $(1, 1, 0) .$ then which of the following statement(s) is/are correct ?

The equation of plane

P_{1}

is :

x + y = 2

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The equation of plane

P_{1}

is :

x - y = 2

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The equation of plane

P_{2}

is:

x - z = 0

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The equation of plane

P_{2}

is:

y - z = 0

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Solution

The correct option is D The equation of plane $P_{2}$ is: $y - z = 0$
Given : line $L_{1} : \frac{x - 1}{1} = \frac{y - 1}{- 1} = \frac{z - 1}{- 1}$ passes through $(1, 1, 1), d . r^{'} s = (1, - 1, - 1)$
Equation of plane $P_{1}$ containing $L_{1}$ and having normal $d . r^{'} s = (1, 1, 0)$ is given by,
$1 (x - 1) + 1 (y - 1) + 0 = 0 \Rightarrow x + y = 2 \dots (i)$
and equation of plane $P_{2}$ containing $L_{1}$ and passes through $(0, 1, 1)$ is given by,
$∣ ∣ ∣ ∣ \begin{matrix} x - 1 & y - 1 & z - 1 1 - 0 & 1 - 1 & 1 - 1 1 & - 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ = 0$
$\Rightarrow y - z = 0$

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Similar questions

Q. In

R^{3}

, let

P_{1}

be the plane which contains the line

L_{1} : \to r =^i +^j +^k + λ (^i -^j -^k)

and

P_{2}

be the another plane which contains the line

L_{1}

and a point with position vector

^j

. If the vector

(^i +^j)

is normal to

P_{1}

, then which of the following is (are) true?

Q. Let a plane

P_{1}

containing the line

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

and another plane

P_{2}

containing the line

L_{1}

and passses through the point

(0, 1, 1) .

If d.r's of normal to the plane

P_{1}

are

1, 1, 0,

then the acute angle between the planes

P_{1}

and

P_{2}

is equal to

Q. 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}

, and

P_{1}

and

P_{2}

, respectively.
STATEMENT-

1

: At least two of the lines

L_{1}, L_{2}

and

L_{3}

are non-parallel.
and
STATEMENT -

2

: The three planes do not have a common point.

Consider the lines $L_{1}$ : $\frac{x - 1}{2} = \frac{y}{- 1} = \frac{z + 3}{1}, L_{2}$ : $\frac{x - 4}{1} = \frac{y + 3}{1} = \frac{z + 3}{2}$ and the planes $P_{1}$ : $7 x + y + 2 z = 3, P_{2}$ : $3 x + 5 y - 6 z = 4.$ Let $a x + b y + c z = d$ be the equation of the plane passing through the point of intersection of lines $L_{1}$ and $L_{2}$ , and perpendicular to planes $P_{1}$ and $P_{2}$ .