Before You Begin

Question #41515

Difficulty: 8/10 | 🟢 Free

The Question

Let $I_n = \int_0^1 \frac{x^n}{1+x^2} \, dx$ for $n \ge 1$ . A family of curves is given by the equation $y^2 = 2kx + cx^2$ , where $c$ is an arbitrary constant and $k = 4(I_5 + I_3)$ . Which of the following represents the orthogonal trajectories of this family of curves? (Here $C$ is an arbitrary constant of integration)

Options

A.
$x^2 + 2y^2 \ln|y| = C y^2$
B.
$x^2 - 2y^2 \ln|y| = C y^2$
C.
$y^2 + 2x^2 \ln|x| = C x^2$
D.
$y^2 - 2x^2 \ln|x| = C x^2$

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

Reduction Formulae|Arbitrary Constants vs Parameters|Orthogonal Trajectories

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual distinction between an arbitrary constant of a family of curves and a fixed parameter defined by a reduction formula.

The trap lies in the elimination step: students often reflexively differentiate to eliminate all constants.

If the reduction constant $k$ is mistakenly treated as the arbitrary constant to be eliminated (instead of $c$ ), it leads to an entirely incorrect differential equation ( $2xyy' - y^2 - cx^2 = 0$ ).

By forcing the evaluation of $k$ via a reduction formula first, the problem ensures $k$ is recognized as a fixed numerical parameter ( $k=1$ ) that must remain in the differential equation, while $c$ is eliminated to find the correct orthogonal trajectory.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Reduction Formulae
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free