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Equation of Line: Symmetrical Form

Consider thre...

Question

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.

Statement-1 is True, Statement -2 is True; Statement-2 is a correct explanation for Statement-1

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Statement -1 is True, Statement -2 is True; Statement-2 is NOT a correct explanation for Statement-1

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Statement -1 is True, Statement -2 is False

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Statement -1 is False, Statement -2 is True

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Solution

The correct option is D Statement -1 is False, Statement -2 is True
Given three planes are
$P_{1} : x - y + z = 1$ ... $(1)$
$P_{1} : x + y - z = - 1$ .... $(2)$
and $P_{1} : x - 3 y + 3 z = 2$ ... $(3)$
Solving Eqs. $(1)$ and $(2)$ , we have
$x = 0, z = 1 + y$
which does not satisfy Eq. $(3)$
As, $x - 3 y + 3 z = 0 - 3 y + 3 (1 + y) = 3 (\neq 2)$
$∴$ Statement II is true.
Nest,since we know that direction ratio's of line of intersection of planes $a_{1} x + b_{1} y + c_{1} z + d_{1} = 0$ and
$a_{2} x + b_{2} y + c_{2} z + d_{2} = 0$
$b_{1} c_{2} - b_{2} c_{1}, c_{1} a_{2} - a_{1} c_{2}, a_{1} b_{1}$
Using above result, we get
Direction ratio's of lines $L_{1}, L_{2}$ and $L_{3}$ are $o, 2, 2; 0, - 4, - 4; 0, - 2, - 2$ respectively.
$\Rightarrow$ All the three lines $L_{1}, L_{2}$ and $L_{3}$ are parallel pairwise.
$∴$ Statement I is false.

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Similar questions

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.

Q. Statement 1: If

P_{1} (x_{1}, y_{1})

and

P_{2} (x_{2}, y_{2})

be the images of point

P (x, y)

about lines

L_{1} \equiv a x + b y + c

and

L_{2} \equiv b x - a y + c^{'} = 0

respectively, then the line joining points

P_{1}

and

P_{2}

always passes through point of intersection of lines

L_{1}

and

L_{2}

Statement 2: Lines

L_{1}

and

L_{2}

are perpendicular

Q. Lines

L_{1} : y - x = 0

and

L_{2} : 2 x + y = 0

intersect the line

L_{3} : y + 2 = 0

P

and

Q

, respectively. The bisector of the acute angle between

L_{1}

and

L_{2}

intersects

L_{3}

at R.

STATEMENT-1 : The ratio

P R : R Q

equals

2 \sqrt{2} : \sqrt{5} .

because

STATEMENT-2: In any triangle, bisector of an angle divides the triangle into two similar triangles.

Q. Assertion :Consider the three planes

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

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

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

Let

L_{1}, L_{2}

and

L_{3}

be the lines of intersection of the planes

P_{1} P_{2}

and

P_{2} P_{3}

and

P_{1} P_{3}

respectively.

At least two of the lines and are non-parallel

Reason: The three planes do not have a common point.

Q. Statement

- 1 :

0 < x < y \Rightarrow {log}_{a} x > {log}_{a} y

, where

a > 1

.
Statement

- 2 :

When

a > 1, {log}_{a} x

is an increasing function.

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