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Question #40760

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_1$ be the curve given by $y = \frac{x^3 - 4x^2 + x + 6}{x - 2}$ and $C_2$ be the line $y = -3$ . Let $P$ be the valid point of intersection of $C_1$ and $C_2$ . If $L$ is the angle bisector of the tangents to $C_1$ and $C_2$ at $P$ such that $L$ has a positive slope, then the $x$ -intercept of $L$ is:

Options

A.
$\frac{3(\sqrt{5} - 1)}{2}$
B.
$\frac{3\sqrt{5} + 7}{2}$
C.
$\frac{3(\sqrt{5} + 1)}{2}$
D.
$\frac{7 - 3\sqrt{5}}{2}$

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

Angle Bisector Logic|Tangent/Normal Slope Interplay|Wavy Curve (Method of Intervals)

Why This Question Is Tricky

Common Mistakes

This question tests the student's vigilance regarding domain constraints during algebraic manipulation.

By equating the curves, students find two apparent intersection points ( $x=0$ and $x=2$ ).

The trap is that $x=2$ is a removable discontinuity and not in the domain of $C_1$ .

Students who blindly apply calculus without verifying the existence condition will calculate the tangent at the phantom point $P(2, -3)$ , leading to the incorrect bisector slope and ultimately the trap $x$ -intercept in Option B.

Quick Facts About This Question

Question Information

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