The points on the curve 5 x^2-6 x y+5 y^2=4 that are the...

The points on the curve $5 x^{2}-6 x y+5 y^{2}=4$ that are the nearest the origin are
(a) $(1 / 2,-1 / 2),(-1 / 2,1 / 2)$
(b) $(0,2 / \sqrt{5}),(0,-2 / \sqrt{5})$
(c) $(2 \sqrt{5}, 0),(-2 / \sqrt{5}, 0)$
(d) $(2 / \sqrt{3}, 0),\left(\frac{2}{\sqrt{5}}, 1\right)$

Key Concepts

Constrained Optimization

Constrained optimization involves finding the minimum or maximum of a function subject to one or more constraints. In many problems, this requires optimizing a function (often representing cost, distance, or energy) while ensuring that the solution lies on a curve, surface, or manifold defined by the constraint equations. This concept is fundamental in various fields, including economics, engineering, and physics.

Lagrange Multipliers

The method of Lagrange multipliers is a powerful tool for solving constrained optimization problems. It involves introducing auxiliary variables (known as Lagrange multipliers) to transform a constrained problem into an unconstrained problem by combining the objective function and the constraints. This approach identifies critical points where the gradient of the function is parallel to the gradient of the constraint, which may correspond to the minimum or maximum of the original function.

Quadratic Forms

Quadratic forms are polynomial expressions in which each term is of degree two. They are widely used to represent conic sections such as ellipses, hyperbolas, and parabolas. In optimization problems, quadratic forms often define the shape of the constraint curve or surface, and understanding their properties can simplify the process of finding extrema, especially when combined with techniques like completing the square or coordinate rotation.

Conic Sections

Conic sections arise from the intersection of a plane with a cone, and they are described by quadratic equations in two variables. The study of conic sections, including ellipses, hyperbolas, and parabolas, is central to understanding various geometric and analytic properties. In optimization contexts, determining distances to these curves (such as the shortest distance from a point to an ellipse) requires the application of techniques from both algebra and calculus, making them an important component of geometric optimization problems.

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