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Question #74428

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_1$ be the hyperbola defined by the position vector $\vec{r}(\theta) = a \sec \theta \hat{i} + a \tan \theta \hat{j}$ for $a > 0$ . At $\theta = \pi/4$ , let $\vec{T}$ be the tangent vector to $C_1$ . A normal vector to $C_1$ at this point is defined by the vector triple product $\vec{N}_1 = \vec{T} \times (\vec{r} \times \vec{T})$ .

Let $C_2$ be the curve defined by $y = \sqrt{2} \ln(x^2 - b^2)$ , where $b > 0$ . Let $S$ be the set of all valid $x$ -coordinates on $C_2$ at which the normal vector to $C_2$ is parallel to $\vec{N}_1$ . Find the sum of the elements in $S$ .

Options

A.
2
B.
$1 + \sqrt{1 + b^2}$
C.
$1 - \sqrt{1 + b^2}$
D.
$2\sqrt{1 + b^2}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Vector Triple Product Expansion|Normal to Conic|Logarithmic Differentiation Domain|Slope Sensitivity

Why This Question Is Tricky

Examiner Levers

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the intersection of vector geometry and calculus domains.

Students must first correctly expand the Vector Triple Product to find the normal slope, leading to the quadratic equation $x^2 - 2x - b^2 = 0$ .

The critical trap lies in the domain of logarithmic differentiation: $y = \sqrt{2} \ln(x^2 - b^2)$ strictly requires $|x| > b$ .

Students who blindly sum the roots using Vieta's formulas will get 2 (Option A), failing to reject the extraneous root $x = 1 - \sqrt{1 + b^2}$ which violates the domain constraint.

This simulates the classic JEE Advanced mechanism of 'ghost solutions' generated by algebraic manipulation of restricted functions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Vector Triple Product Expansion
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free