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Question #40865

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$ , $\vec{b} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ , and $\vec{c} = 3\hat{i} + \lambda\hat{j} + 5\hat{k}$ be three coplanar vectors. A point $P$ with position vector $\vec{p} = x\hat{i} + y\hat{j} + z\hat{k}$ divides the line segment joining $A(0, 4, 2)$ and $B(2, 0, 4)$ internally in the ratio $k:1$ . If $\vec{p}$ satisfies the system of equations $\vec{p} \cdot \vec{a} = 14$ $\vec{p} \cdot \vec{b} = 20$ $\vec{p} \cdot \vec{c} = \mu$ and the system is consistent, then the value of $\mu$ and the set of all possible values for $k$ are:

Options

A.
$\mu = 26, \; k = 1$
B.
$\mu = 26, \; k \in (0, \infty)$
C.
$\mu = 24, \; k = 2$
D.
$\mu = 24, \; k \in (0, \infty)$

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

Ratio of Division (k:1)|Collinearity vs Coplanarity|Cramer's Rule Constraints

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of 3D vector geometry with systems of linear equations (Cramer's Rule).

The primary trap lies in the assumption that a system of plane equations will yield a unique point $P$ , and thus a unique ratio $k$ .

By forcing the scalar triple product to zero, the system becomes dependent ( $\Delta = 0$ ).

Students must use the consistency condition ( $\Delta_x = 0$ ) to find $\mu$ .

The 'Lethal Collision' occurs when substituting the section formula coordinates into the plane equations results in an identity ( $14=14$ ).

Students often misinterpret this as a calculation error or assume a default value like $k=1$ , failing to realize that the entire line segment lies on the planes' intersection, making any internal ratio $k \in (0, \infty)$ valid.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Ratio of Division (k:1)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free