Question

The equation \(x^{2}+y^{2}=4\) represents a circle with center at (0,0) and with a radius of 2 (a) Interpret geometrically the set \(\left.f(x, y) | x^{2}+y^{2} \leq 4\right\\}\) (b) Is this set convex?

Short answer

(a) It represents the interior and boundary of a circle with radius 2. (b) Yes, it is convex.

Step-by-step solution

Interpreting the given set

The given set \(\{ (x, y)\,|\, x^2 + y^2 \leq 4 \}\) represents all the points \((x, y)\) in the coordinate plane that satisfy \(x^2 + y^2 \leq 4\). This inequality describes the region inside or on a circle centered at the origin \((0,0)\) with a radius of 2. Therefore, geometrically, the set includes all points on and inside a circle with a radius of 2 and center at the origin.

Definition of Convex Sets

A set is convex if for any two points within the set, the line segment connecting them lies entirely within the set. In other words, if \((x_1, y_1)\) and \((x_2, y_2)\) are any two points in the set, then every point \((1-t)\cdot(x_1, y_1) + t\cdot(x_2, y_2)\) for \(0 \leq t \leq 1\) should also be in the set.

Checking Convexity of the Set

Consider any two points \((x_1, y_1)\) and \((x_2, y_2)\) inside the circle \(x^2 + y^2 \leq 4\). The line segment connecting these points can be expressed as \((1-t)\cdot(x_1, y_1) + t\cdot(x_2, y_2)\) where \(0 \leq t \leq 1\). For each \(t\), the expression defines points that must satisfy \(x^2 + y^2 \leq 4\) because they are linear combinations of points that already satisfy the inequality. As the interior of a circle is a well-known convex shape, by the definition of convexity confirmed through interpolation of any two points within it, we conclude that the entire region including boundary defined by \(x^2 + y^2 \leq 4\) is convex.

Key concepts

Geometric Interpretation

The equation given, \(x^2 + y^2 = 4\), is a fundamental form in coordinate geometry, representing a circle centered at the origin \((0,0)\) with a radius of 2. When interpreting this geometrically, we consider the inequality \(x^2 + y^2 \leq 4\). This inequality expands our consideration from merely the boundary of the circle to include all points within and on the circle. Visualize the circle: - **Boundary**: Defined by points \((x, y)\) where \(x^2 + y^2 = 4\). Imagine a solid line outlining the circle, showing where the boundary lies.- **Interior**: Encompasses all points inside the circle where \(x^2 + y^2 < 4\). This is the entire area enclosed by the circle.Together, the boundary and interior define a disk with a center at \((0,0)\) and a radius of 2 units. Every point on this disk satisfies the inequality, thus making it a complete geometric set of points within the defined limit.

Inequalities

In mathematics, inequalities describe the relationship between two values when they are not exactly equal. The inequality \(x^2 + y^2 \leq 4\) tells us that for any point on the plane, its distance from the origin should be less than or equal to 2.Understanding through steps:- **Comparison**: The inequality is a comparison between the variable expression \(x^2 + y^2\) and the constant 4.- **Boundary Condition**: When \(x^2 + y^2 = 4\), the points lie exactly on the circle's boundary.- **Interior Condition**: When \(x^2 + y^2 < 4\), the points exist inside the circle.Inequalities allow us to encompass a range of solutions, expressing a broad set of possible answers within certain bounds. In the context of the circle, it determines the entire set of solutions residing on or within the circle's boundary.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves studying geometric figures using a coordinate system. For our exercise, the focus is on a circle within the standard Cartesian coordinate plane, which is a two-dimensional system filled with points defined by \(x\) and \(y\) coordinates.Key components include:- **Origin**: The fixed point \((0,0)\) where the circle is centered.- **Circle Equation**: \(x^2 + y^2 = r^2\), where \(r\) is the radius.- **Radius**: Distance from the center to any point on the boundary; here, it is 2.Through coordinate geometry, we translate geometric problems into algebraic equations, making them easier to handle. Transforming the problem of a circle into \(x^2 + y^2 \leq 4\) allows us to use algebraic techniques to solve and understand properties like convexity and boundary conditions. This approach links algebra with geometry, providing a powerful tool to visualize and interpret mathematical concepts.