Geometric Regions

date2026-07-15document_iddoc_df79ff50f8699fb962163b93e63165bbdescription方程式を領域の境界、不等式を境界の一方を指定する条件として解釈し、半平面、円の内部、共通部分を扱う。prerequisites図形と方程式の基本 / 不等式の基本 / 座標平面の基本type講義content_typelecturestatusactive

mathgeometryhighschoollecture

1Introduction

This lecture interprets inequalities in two variables as conditions on points in the plane. For linear expressions and squared distances from a center, it explains how an equation specifies a boundary and an inequality selects one side of that boundary.

2Regions and boundaries

A region is a set of points in the plane satisfying a condition. Its boundary is the set separating points inside the region from points outside it. For a general inequality, the solution set of the corresponding equation need not be exactly the boundary. The cases below are restricted to linear expressions and circles, for which the boundary can be identified precisely.

3Half-planes determined by a line

Let $(a, b) \neq (0, 0)$ and define

L (x, y) = a x + b y + c .

If $b \neq 0$ , then $L (x, y) = b {y - (- a x - c) / b}$ . Its sign is constant on each side of the line $y = (- a x - c) / b$ and reverses upon crossing the line. If $b = 0$ , then $a \neq 0$ and $L (x, y) = a (x + c / a)$ , which gives the same conclusion on the two sides of the vertical line $x = - c / a$ . Thus $L (x, y) = 0$ divides the plane into two half-planes, and $L (x, y) \geq 0$ represents one half-plane together with its boundary line.

Choose a test point that does not lie on the boundary and substitute its coordinates into $L$ . The resulting sign identifies the half-plane satisfying the inequality. When the origin is not on the boundary, $(0, 0)$ is often a convenient test point.

3.1Example

The boundary of

x + y \leq 2

is $x + y = 2$ . Substitution of $(0, 0)$ gives $0 \leq 2$ , so the required region is the side containing the origin, including the boundary line.

4Regions determined by a circle

The quantity

D (x, y) = (x - a)^{2} + (y - b)^{2}

is the squared distance from $(x, y)$ to the center $(a, b)$ . Therefore $D (x, y) < r^{2}$ describes the interior of the circle, $D (x, y) = r^{2}$ its circumference, and $D (x, y) > r^{2}$ its exterior. The inequality $D (x, y) \leq r^{2}$ describes the closed disk: the union of the interior and circumference.

5Intersections of conditions

The region satisfying several inequalities simultaneously is the intersection of their individual regions. Identify every boundary, determine the side selected by each condition, and then retain their common part.

6Summary

In the linear and circular cases treated here, the corresponding equation specifies the boundary of the region.
Substitution of a test point determines the half-plane selected by a linear inequality.
A circular inequality compares the distance from the center with the radius.
Simultaneous conditions describe the intersection of their regions.