P1 and P2 are planes passing through...

MathematicsEquation of a PlaneJEE Advanced 2004Moderate

$P_{1}$ and $P_{2}$ are planes passing through origin. $L_{1}$ and $L_{2}$ are two line on $P_{1}$ and $P_{2}$ respectively such that their intersection is origin. Show that there exists points $A, B, C$ whose permutation $A^{'}, B^{'}, C^{'}$ can be chosen such that (i) $A$ is on $L_{1}, B$ on $P_{1}$ but not on $L_{1}$ and $C$ not on $P_{1}$ (ii) $A^{'}$ is on $L_{2}, B^{'}$ on $P_{2}$ but not on $L_{2}$ and $C^{'}$ not on $P_{2}$ .

Visualized Solution (Hindi)

Visualizing the Planes .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P_{1} and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P_{2}

Let $P_{1}$ and $P_{2}$ be two planes passing through the origin $O (0, 0, 0)$ .
The intersection of $P_{1}$ and $P_{2}$ is a line passing through the origin.
Let this intersection line be denoted as $L_{in t} = P_{1} \cap P_{2}$ .

Defining Lines .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L_{1} and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L_{2}

Line $L_{1} \subset P_{1}$ and line $L_{2} \subset P_{2}$ .
Given: $L_{1} \cap L_{2} = {O}$ .
This implies $L_{1}$ and $L_{2}$ are distinct lines and neither is identical to the intersection line $L_{in t}$ (except at the origin).

The Strategic Choice of Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Choose point $B$ such that $B \in L_{in t}$ and $B \neq = O$ .
Since $L_{in t} \subset P_{1}$ , then $B \in P_{1}$ .
Since $L_{1} \cap L_{in t} = {O}$ and $B \neq = O$ , then $B \in / L_{1}$ .

Selecting Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A on .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L_{1}

Choose point $A$ such that $A \in L_{1}$ and $A \neq = O$ .
This satisfies the condition: $A$ is on $L_{1}$ .

Selecting Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C outside .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P_{1}

Choose point $C$ such that $C \in L_{2}$ and $C \neq = O$ .
Since $L_{2} \cap P_{1} = {O}$ and $C \neq = O$ , then $C \in / P_{1}$ .
Summary for $(A, B, C)$ : $A \in L_{1}$ , $B \in P_{1} ∖ L_{1}$ , and $C \in / P_{1}$ .

The Permutation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A^{'}, B^{'}, C^{'}

Define the permutation $(A^{'}, B^{'}, C^{'})$ as $(C, B, A)$ .
This means: $A^{'} = C$ , $B^{'} = B$ , and $C^{'} = A$ .

Verifying the Second Condition

1. $A^{'} = C \in L_{2}$ (By construction).
2. $B^{'} = B \in L_{in t} \subset P_{2}$ and $B \in / L_{2}$ (Since $L_{in t} \cap L_{2} = {O}$ ).
3. $C^{'} = A \in L_{1}$ and $L_{1} \cap P_{2} = {O}$ , so $A \in / P_{2}$ .
All conditions for $(A^{'}, B^{'}, C^{'})$ are satisfied.

Key Takeaways & Conclusion

Key Takeaway: Strategic selection of points on the intersection line $P_{1} \cap P_{2}$ allows them to satisfy conditions for both planes.
Symmetry: The problem relies on the symmetric roles of $L_{1}$ and $L_{2}$ relative to the intersection of the planes.
Next Challenge: What if the lines $L_{1}$ and $L_{2}$ were skew and did not pass through the origin? How would the existence proof change?

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Visualized Solution (Hindi)

Visualizing the Planes $P_{1}$ and $P_{2}$

Defining Lines $L_{1}$ and $L_{2}$

The Strategic Choice of Point $B$

Selecting Point $A$ on $L_{1}$

Selecting Point $C$ outside $P_{1}$

The Permutation $A^{'}, B^{'}, C^{'}$

Verifying the Second Condition

Key Takeaways & Conclusion

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Visualized Solution (Hindi)

Visualizing the Planes .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P1​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P2​

Defining Lines .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L1​ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L2​

The Strategic Choice of Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } B

Selecting Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A on .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } L1​

Selecting Point .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C outside .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } P1​

The Permutation .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A′,B′,C′

Verifying the Second Condition

Key Takeaways & Conclusion

Conceptually Similar Problems

List-I

List-II

List-I

List-II

Visualizing the Planes $P_{1}$ and $P_{2}$

Defining Lines $L_{1}$ and $L_{2}$

The Strategic Choice of Point $B$

Selecting Point $A$ on $L_{1}$

Selecting Point $C$ outside $P_{1}$

The Permutation $A^{'}, B^{'}, C^{'}$