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Question #40882

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be a variable line passing through the origin such that its direction cosines $(l, m, n)$ satisfy $l \ge m \ge 0$ and $n = \frac{1}{\sqrt{2}}$ . Let $Q$ be the foot of the perpendicular drawn from the point $P(3, 4, 5)$ to the line $L$ . The maximum value of the distance $OQ$ , where $O$ is the origin, is:

Options

A.
$\frac{7 + 5\sqrt{2}}{2}$
B.
$5\sqrt{2}$
C.
$\frac{7\sqrt{2} + 5}{2}$
D.
$4\sqrt{2}$

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

Projection vs. Coordinate|Direction Cosines constraint|Monotonicity and Boundedness

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to express the projection of a point onto a variable line and optimize it under direction cosine constraints.

The core trap lies in the monotonicity and boundedness of the resulting trigonometric function.

Students typically use the unconstrained maximum of $a\cos\theta + b\sin\theta$ , which is $\sqrt{a^2+b^2}$ , yielding the incorrect answer $5\sqrt{2}$ .

They fail to realize that the geometric constraint $l \ge m \ge 0$ restricts the domain of $\theta$ to $[0, \pi/4]$ .

Within this restricted domain, the critical point $\theta = \arctan(4/3)$ is unreachable, making the function strictly monotonic and forcing the maximum to occur at the boundary $\theta = \pi/4$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Projection vs. Coordinate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free