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Question #41402

Difficulty: 8/10 | 🟢 Free

The Question

A particle $P(x,y,z)$ moves along the curve of intersection of the sphere $x^2 + y^2 + z^2 = 14$ and the surface $xyz = 6$ . At the instant when the particle is at the point $(1, 2, 3)$ , its $x$ -coordinate is increasing at a rate of $5$ units/sec. Let $L$ be the line of intersection of the planes $x + y + z = 3$ and $x - y + z = 1$ . What is the rate of change of the distance from the particle $P$ to the line $L$ at this instant?

Options

A.
$-\frac{2}{\sqrt{3}}$
B.
$\frac{2}{\sqrt{5}}$
C.
$2\sqrt{3}$
D.
$-\frac{4}{\sqrt{3}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Implicit Differentiation Rate|Line-Plane intersection logic|Equation of plane through intersection|Angle between two planes

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to combine implicit differentiation of 3D trajectories with the vector geometry of line-plane intersections.

The primary trap lies in calculating the rate of change of the distance between the moving particle and the intersection line.

Students often mistakenly calculate the rate of change of the distance to a fixed point on the line (yielding $2/\sqrt{5}$ ) rather than correctly projecting the position vector onto the line's direction and subtracting this component.

This requires recognizing that the distance formula $D^2 = |\vec{AP}|^2 - (\vec{AP} \cdot \hat{u})^2$ must be differentiated entirely, accounting for both the radial velocity and the velocity component parallel to the line.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Implicit Differentiation Rate
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free