Before You Begin

Question #40839

Difficulty: 8/10 | 🟢 Free

The Question

Let $H_x$ be a family of hyperbolas defined parametrically for each real $x \ge 0$ by: $X(\theta) = 2^{[x]} \sec \theta, \quad Y(\theta) = 3^{[x]} \tan \theta$ where $\theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $[ \cdot ]$ denotes the greatest integer function. Let $L_1(x)$ and $L_2(x)$ be the asymptotes of $H_x$ , and $P_x$ be the point on $H_x$ corresponding to the parameter $\theta = \frac{\pi}{4}$ . If $d_1(x)$ and $d_2(x)$ are the perpendicular distances from $P_x$ to $L_1(x)$ and $L_2(x)$ respectively, evaluate the integral: $I = \int_0^\infty \frac{1}{d_1(x) d_2(x)} \, dx$

Options

A.
$\frac{59}{24}$
B.
$\frac{25}{12}$
C.
$\frac{61}{24}$
D.
$\frac{17}{8}$

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

Asymptote Geometry|Parametric Form Manipulation|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate parametric coordinate geometry with piecewise functional analysis.

The trap is embedded by setting the asymptote coefficients as a function of $[x]$ , causing the asymptote slopes to 'jump' discontinuously at every integer.

Students who attempt to brute-force the distance calculations by plugging in the coordinates of $P_x$ will face a nightmare of absolute values and piecewise intervals.

The key to bypassing this trap is recognizing the geometric invariant: the product of perpendicular distances from any point on a hyperbola to its asymptotes is always $a^2b^2/(a^2+b^2)$ , completely independent of the parametric angle $\theta$ .

This reduces the complex geometry problem into a clean evaluation of two infinite geometric series.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Asymptote Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free