Introduction

This article aims to provide a detailed approach to solving a mathematical problem: finding the point on a conic section that has the minimum distance from the origin. We will explore a step-by-step process in solving this problem, focusing on the specific conic section given and utilizing techniques such as partial derivatives and translations. By the end, we will have a clear understanding of how to apply these concepts to arrive at the minimum distance.

Problem Statement

We are given the conic section equation: x^2 - y^2 - 6x - 4y - 5 0 Our goal is to find the point on this curve that is closest to the origin (0,0), and subsequently, the minimum value of the distance squared, x^2 y^2.

Step-by-Step Solution

Step 1: Analyzing the Conic Section

The given conic section is a hyperbolic type. To find its center, we vanill the partial derivatives: 2x - 6 0 -2y - 4 0 Solving these equations, we find the coordinates of the center to be (3, -2).

Step 2: Translation

To simplify the equation, we perform a coordinate transformation:

Substituting these into the original equation, we get:

X^2 - Y^2 - 0 0

This simplifies to:

X - Y 0 and XY 0

In the original reference system, these lines are:

x - y - 5 0 and xy - 1 0

Step 3: Finding Minimum Distance

Next, we find the squared distance from the origin to each line and determine which is the shortest. For the first line:

Distance frac{|0 - 0 - 5|}{sqrt{1^2 1^2}} frac{5}{sqrt{2}}

And for the second line:

Distance frac{|0 times 0 - 1|}{sqrt{1^2 1^2}} frac{1}{sqrt{2}}

The second line has the shortest distance, leading us to a minimum squared distance of frac{1}{2}.

Verification Using Lagrange Multipliers

Another method to solve this problem is by using the Lagrange Multiplier technique. We introduce a function f(x, y) x^2 y^2 (to minimize) and a constraint g(x, y) x^2 - y^2 - 6x - 4y - 5 0.

We define:

F(x, y, lambda) f(x, y) - lambda g(x, y)

Substituting and differentiating, we have:

frac{partial F}{partial x} 2x - 2lambda x - 6 0

frac{partial F}{partial y} 2y 2lambda y - 4 0

Solving these equations:

lambda -x/(3x)

lambda y/(y - 2)

Equating the two expressions for lambda, we get:

y/(y - 2) -x/(3x)

Solving for x and y, we find the points that satisfy the equation.

Conclusion

The minimum value of the squared distance from the origin to the given conic section is frac{1}{2}. This solution demonstrates a combination of geometric insight and algebraic techniques to solve a complex optimization problem. Understanding these methods is invaluable for tackling similar problems in calculus and geometry.