JEE Advanced · Mathematics · 31. 3D Geometry

Let $P_{1}$ and $P_{2}$ be two planes given by $P_{1} : 10 x + 15 y + 12 z - 60 = 0,$ $P_{2} : - 2 x + 5 y + 4 z - 20 = 0$ . Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?

A $\frac{x - 1}{0} = \frac{y - 1}{0} = \frac{z - 1}{5}$
B $\frac{x - 6}{- 5} = \frac{y}{2} = \frac{z}{3}$
C $\frac{x}{- 2} = \frac{y - 4}{5} = \frac{z}{4}$
D $\frac{x}{1} = \frac{y - 4}{- 2} = \frac{z}{3}$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{x - 1}{0} = \frac{y - 1}{0} = \frac{z - 1}{5}$

Step-by-step Solution

Detailed explanation

Given,

P_{1}

and

P_{2}

be two planes given by

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let P1 and P2 be two planes given by P1:10x+15y+12z-60=0, P2:-2x+5y+4z-20=0. Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on P1 and P2?

Step-by-step Solution

Let $P_{1}$ and $P_{2}$ be two planes given by $P_{1} : 10 x + 15 y + 12 z - 60 = 0,$ $P_{2} : - 2 x + 5 y + 4 z - 20 = 0$ . Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_{1}$ and $P_{2}$ ?