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Family of Planes Passing through the Intersection of Two Planes

Assertion :Co...

Question

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.

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is D Assertion is incorrect but Reason is correct
Given three planes are
$P_{1} : x - y + z = 1$ ...(i)
$P_{2} : x + y - z = - 1$ ...(ii)
and $P_{3} : x - 3 y + 3 z = 2$ ...(iii)
Solving Eqs.(I) and (ii), we have
$x = 0, z = 1 + y$
which does not satisfy Eq.(iii)
As, $x - 3 y + 3 z = 0 - 3 y + 3 (1 + y) = 3 (\neq 2)$
Therefore Statement II is true.
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 $a_{2} x + b_{2} y + c_{2} z + d_{2} = 0$
$b_{1} c_{2} - b_{2} c_{1}, c_{1} a_{2} - a_{1} c_{2}, a_{1} b_{2} - a_{2} b_{1}$
Using above result ,
Direction ratio's of lines $L_{1}, L_{2}$ and $L_{3}$ are $0, 2, 2,; 0, - 4, - 4; 0, - 2, - 2$ 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.

Assertion: All the points (1, 0), (-1, 0), (2, 0) and (5, 0) lie on the x – axis.
Reason: Equation of the x-axis is y = 0

Choose the correct option.

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

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and

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1

: At least two of the lines

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and

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and
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Assertion :Consider the three planes P1:x−y+z=1 P2:x+y−z=1 P3:x−3y+3z=1 Let L1,L2 and L3 be the lines of intersection of the planes P1P2 and P2P3 and P1P3 respectively. At least two of the lines and are non-parallel Reason: The three planes do not have a common point.