COMEDK · Maths · 34. Three Dimensional Geometry

If the straight lines \(\dfrac{x-2}{1}=\dfrac{y-3}{1}=\dfrac{z-4}{-t}\) and \(\dfrac{x-1}{t}=\dfrac{y-4}{2}=\dfrac{z-5}{1}\) are intersecting then \(t\) can have

A Any number of values
B Exactly one value
C Exactly three values
D Exactly two values

Verified Solution

Answer & Solution

Correct Answer

(D) Exactly two values

Step-by-step Solution

Detailed explanation

The given lines are \(L_1: \dfrac{x-2}{1} = \dfrac{y-3}{1} = \dfrac{z-4}{-t} = \lambda\) and \(L_2: \dfrac{x-1}{t} = \dfrac{y-4}{2} = \dfrac{z-5}{1} = \mu\). The general points on the lines are \(P_1 = (2+\lambda, 3+\lambda, 4-t\lambda)\) and \(P_2 = (1+t\mu, 4+2\mu, 5+\mu)\).…

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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