Before You Begin

Question #40751

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_i: a_i x + b_i y + c_i = 0$ ( $i=1,2,3$ ) be three distinct, non-parallel lines in the $xy$ -plane that are concurrent at the point $P(\alpha, \beta)$ , where $\alpha \beta \neq 0$ . A new system of lines is defined as $L_i': (a_i + k c_i)x + (b_i + m c_i)y + c_i = 0$ for $i=1,2,3$ , where $k, m \in \mathbb{R}$ . If the system of lines $L_i'$ is inconsistent, which of the following conditions must hold true?

Options

A.
$k \alpha + m \beta = 1$
B.
$k \alpha - m \beta = 1$
C.
$k \beta + m \alpha = 1$
D.
$k \alpha + m \beta = 0$

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

Concurrent Lines Condition|Row/Column Exchange Parity|Inconsistency Detection

Why This Question Is Tricky

Common Mistakes

This question tests the deep connection between determinant properties, linear dependence, and the geometric consistency of linear systems.

A common trap is assuming that $\Delta = 0$ strictly implies concurrency.

Here, column operations guarantee that the new system's determinant is also zero.

Students who blindly rely on the determinant condition will fail to distinguish between a concurrent system and an inconsistent one.

The question forces the realization that an inconsistent system with $\Delta = 0$ must consist of parallel lines, translating this geometric constraint into a linear dependence relation that must align with the original null space spanned by the concurrency point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Concurrent Lines Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free