Before You Begin

Question #41629

Difficulty: 8/10 | 🟢 Free

The Question

Let $y = f(x)$ be the general solution to the homogeneous differential equation $\frac{dy}{dx} = \frac{y^3 + x^2 y}{x^3 + x y^2}$ . Consider the family of planes $\Pi_c$ in $\mathbb{R}^3$ given by the equation $y = f(x)$ , where $c \in \mathbb{R}$ is the constant of integration. Let $I_c$ be the image of the point $A(3, 4, 0)$ in the plane $\Pi_c$ . As $c$ varies over all real numbers, $I_c$ traces a curve $\Gamma$ . Find the minimum distance of the curve $\Gamma$ from the point $P(8, 15, 12)$ .

Options

A.
$12\sqrt{2}$
B.
$13\sqrt{2}$
C.
15
D.
$10\sqrt{3}$

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

Image of a point in a plane|Homogeneous Function Homogeneity|Geometric Interpretation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to geometrically interpret the constant of integration in a 3D context.

The lethal trap is attempting to solve the optimization problem purely algebraically by finding the coordinates of the image point $I_c$ as a function of $c$ and differentiating the complex distance formula.

While mathematically valid, this algebraic brute-force approach is highly prone to calculation errors and consumes excessive time.

The intended 'Advanced' logical leap is recognizing that reflecting a fixed point across a family of planes intersecting at a common line (the z-axis) generates a circular locus.

By shifting from algebraic optimization to geometric locus properties, the problem simplifies into finding the shortest distance from a point to a circle in 3D space.

Quick Facts About This Question

Question Information

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