The number of points of the form (a+1, √(3) a), a ∈Z lying inside the region bounded by x^2+y^2-

step 1 In this problem we have to find the number of points of this form lying inside the region bounde

The number of points of the form (a+1, √(3) a), a ∈Z lying...

Key Concepts

Circle Equations

Circle equations describe the set of all points in the plane that are at a constant distance (the radius) from a fixed point (the center). Understanding these equations is fundamental in coordinate geometry as it allows one to analyze and compare the positions and sizes of circles based on their algebraic representations.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square plus or minus a constant. In the context of circle equations, this technique is essential for rewriting a general quadratic form into the standard form of a circle's equation, thereby clearly revealing its center and radius.

Annular Region

An annular region is the area between two concentric circles. This concept is important in geometry when one needs to determine whether a given point falls within a region defined not by a single circle, but by the space between an inner and outer boundary.

Parametric Representation of Points

Parametric representation involves expressing the coordinates of points in the plane as functions of one or more parameters. This form is useful for describing a collection of points that follow a specific linear or nonlinear relationship, helping to analyze their spatial distribution.

Discrete Lattice Points

Discrete lattice points refer to points in the coordinate plane with integer or specifically defined parameters. In problems involving geometric regions, determining whether such discretely defined points lie inside, outside, or on the boundary of a region is a common task in analytic geometry.

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