The points on the curve 5 x^2-x y+5 y^2=4 where distance...

The points on the curve $5 x^{2}-x y+5 y^{2}=4$ where distance from the origin is max. or min. are
(a) $(\sqrt{2}, \sqrt{2})$
(b) $(-\sqrt{2},-\sqrt{2})$
(c) $\left(\frac{\sqrt{2}}{3},-\frac{\sqrt{2}}{3}\right)$
(d) $\left(-\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3}\right)$

Key Concepts

Quadratic Forms

Quadratic forms are polynomial expressions where the variables are all of degree two, typically represented in matrix form. They play a critical role in many optimization problems, especially when the constraint and the objective function are both quadratic, as they provide a structured way to analyze geometric properties like curvature and orientation of conic sections.

Distance Function

The distance function, particularly when measuring distance from the origin, is typically represented by the Euclidean norm. In optimization problems, working with the square of the distance can simplify calculations as it removes the square root, preserving the locations of maxima or minima while making derivations easier.

Lagrange Multipliers

Lagrange multipliers is a method used in constrained optimization which helps to find the local maxima and minima of a function subject to equality constraints. By introducing a multiplier for each constraint, the technique transforms the problem into finding stationary points of a single function that combines both the objective and the constraint.

Constrained Optimization

Constrained optimization involves optimizing an objective function while satisfying one or more restrictions or constraints. This concept is fundamental in many areas of mathematics and economics and is often tackled using techniques like Lagrange multipliers to find optimal solutions when direct methods are not applicable.

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