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Question #41732

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(1, 4, 5)$ and $B(7, -5, -4)$ be two points in $\mathbb{R}^3$ . A point $P$ divides the line segment $AB$ internally in the ratio $\lambda : 1$ . It is given that the coordinates $(x, y, z)$ of $P$ satisfy the following system of linear equations: $x + y + z = 2$ $2x + 3y - z = 5$ $3x + 4y = -\lambda^2 + 3\lambda + 5$ The sum of all possible valid values of $\lambda$ is:

Options

A.
2
B.
3
C.
$2 + \sqrt{7}$
D.
$1 + \sqrt{7}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Parametric form pitfalls|Section Formula Internals|Inconsistency Detection

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between algebraic substitution and geometric consistency in a system of linear constraints.

The primary trap lies in substituting the parametric coordinates of $P$ directly into the third equation, which yields the roots $\lambda = 2$ and $\lambda = \sqrt{7}$ (ignoring the negative root for internal division).

However, the system itself is inconsistent for $\lambda = \sqrt{7}$ , meaning no such point can exist in the intersection of the planes.

Furthermore, checking the consistency condition $L_1 + L_2 = L_3$ yields $\lambda \in \{1, 2\}$ , but $\lambda = 1$ produces a point that does not lie on the planes.

Students must verify that the point both makes the system consistent and actually lies on the required locus.

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