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Question #40768

Difficulty: 8/10 | 🟢 Free

The Question

Consider the lines $L_1: x + (a-1)y = 1$ , $L_2: ax + 2y = 2$ , and $L_3: a^2 x + 4y = a+3$ , where $a \in \mathbb{R}$ . Let $S$ be the set of all values of $a$ for which the three lines are concurrent at a unique point. The sum of the elements of $S$ is

Options

A.
$4$
B.
$6$
C.
$2$
D.
$8$

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

Family of Lines|System of Linear Equations (Cramer/Inverse)|Singular vs Non-Singular Matrices

Why This Question Is Tricky

Common Mistakes

This question tests the consistency of a system of linear equations and its geometric interpretation as a family of lines.

The trap lies in the condition $\Delta = 0$ , which is necessary but not sufficient for concurrency at a unique point.

Students often blindly solve the determinant equation and sum all its roots (getting 6).

However, a singular matrix can also represent an inconsistent system (parallel lines) or dependent equations without a unique intersection.

Checking the geometric validity of each root is crucial; here, $a=2$ yields parallel/coincident lines, leaving only the irrational roots as valid parameters.

Quick Facts About This Question

Question Information

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