As for the circumference, it is known that all its points are equally distant from the center, this equal distance is called the radius. In comparison with this radius, that is, with the elements that belong to the circle, we can have 3 positions to be studied between a point and a circle.
To study these relative positions let's determine a circle λ of center C(Xc, Yc) and radius r. We will analyze the relative position of any point P with respect to this circle λ.
• Point P inside the circle: this implies that the distance from point P to the center is less than the radius of the circle.
• Point P outside the circle: in this case we have that the distance from point P to the center is greater than the radius
• Point P belongs to the circle: finally, we have the case where the distance from the point P to the center is equal to the radius.
Therefore, when you know the radius of the circle and you want to analyze the relative position of a point to a given circle, just compare the distance from the Point to the center of the circle with the radius value, then you will be able to determine the positions relative. Thus, it is necessary to know how to calculate the distance between two points, this study you can follow in the article
Distance between two Points.
Let's look at some situations to perform this type of analysis regarding the relative positions between a point and a circle.
"Analyze the relative positions between the given points and the circumference λ: (x+1)2 + (y+1)2=9, whose points are: A(-2,2). B (-4.1), D(1.1), E(-4,-1)"
We must obtain two pieces of information needed to perform the calculations, which are the coordinates of the Center of the circumference and radius, from the reduced equation we can easily obtain these two pieces of information: C (-1, -1) and radius 3.
Just calculate the distances from the points to the center and compare with the radius.
Let's look at the graphical representation of the relative positions of these points in relation to the circumference.
See that only with the concept of distance between points it was possible to approach several themes of analytic geometry. The distance between points is present in practically all analytic geometry, if not all of it.
By Gabriel Alessandro de Oliveira
Graduated in Mathematics
Brazil School Team
Source: Brazil School - https://brasilescola.uol.com.br/matematica/posicoes-relativas-entre-um-ponto-uma-circunferencia.htm