Before You Begin

Question #41739

Difficulty: 8/10 | 🟢 Free

The Question

Let $H$ be the rectangular hyperbola $x^2 - y^2 = 1$ in the $xy$ -plane. A ruled surface $S$ is generated by revolving $H$ about the $y$ -axis. A plane $\Pi$ is given by the equation $ex + y + z = e$ , where $e$ is the eccentricity of $H$ . The plane $\Pi$ intersects the surface $S$ in two distinct straight lines $L_1$ and $L_2$ . If $\theta$ is the acute angle between $L_1$ and $L_2$ , then the value of $\cos \theta$ is:

Options

A.
$\frac{1}{3}$
B.
$\frac{3}{\sqrt{17}}$
C.
$\frac{1}{\sqrt{3}}$
D.
$\frac{2}{3}$

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

Line lying in a plane|Rectangular Hyperbola

Why This Question Is Tricky

Common Mistakes

This question tests the geometric interpretation of ruled surfaces and their generators.

The trap (GEOMETRIC_INT_NON_PARAMETRIC) occurs when students treat the hyperbola as a static 2D curve, intersecting the plane $\Pi$ with the $xy$ -plane to find two points $(1,0,0)$ and $(3,-2\sqrt{2},0)$ .

They then incorrectly assume the lines $L_1$ and $L_2$ are simply the 2D tangents to the hyperbola at these points, which gives direction vectors $(0,1,0)$ and $(-2\sqrt{2},3,0)$ , leading to the trap answer $3/\sqrt{17}$ .

The correct approach requires recognizing the 3D nature of the hyperboloid of one sheet and finding its actual 3D generators lying in the plane.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line lying in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free