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Question #41729

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be a line passing through the point $P(1, 2, 3)$ with direction cosines $l, m, n$ . Let $Q(x, y, z)$ be any point on $L$ and $r$ be the distance between $P$ and $Q$ . Suppose that for all points $Q$ on $L$ , the coordinates satisfy the relation $|x - 1| + 2|y - 2| + 2|z - 3| = 3r$ Let $S$ be the set of all such lines $L$ . If a line $L \in S$ intersects the plane $x + 2y + 2z = 20$ at a point $R$ , let $d_L$ be the distance $PR$ . Find the maximum value of $d_L$ as $L$ varies over $S$ .

Options

A.
27
B.
$\frac{27}{7}$
C.
9
D.
3

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Parametric form pitfalls|Direction Cosines constraint|Triangle Inequality Mind Games

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of parametric line constraints with vector inequalities.

The primary trap is that students might assume the relation $|x-1| + 2|y-2| + 2|z-3| = 3r$ is a general property of the distance and fail to realize it acts as a strict constraint on the direction cosines via the Cauchy-Schwarz inequality.

Without determining the specific discrete set of allowed direction cosines, a student might attempt to minimize the denominator $l+2m+2n$ continuously, incorrectly concluding that the distance can approach infinity (since $l+2m+2n$ could theoretically be arbitrarily close to 0 for a general line).

Recognizing the geometric boundary constraints imposed by the direction cosine limitation is crucial.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free