Answer: TRUE.
Step-by-step explanation:
It knows that the distance "d" between any two points is equal to [tex]\sqrt{(x-x_0)^2-(y-y_0)^2}[/tex].
In a circle the distance between a point that belongs to the circumference and the center ([tex]x_0,y_0[/tex]) is always the same. Then:
[tex]\sqrt{(x-x_0)^2-(y-y_0)^2}[/tex] is always equal to a constant called "r".
This is:
[tex]\sqrt{(x-x_0)^2-(y-y_0)^2}=r[/tex]
Square both sides of the equality:
[tex](\sqrt{(x-x_0)^2-(y-y_0)^2})^2=r^2[/tex]
[tex](x-x_0)^2-(y-y_0)^2=r^2[/tex]
Note that we obtain the General equation of a circle.
This prove that if the distance between a given point in the plane and a collection of points is equal, then the equation of a circle is obtained.
Therefore, the statement "A circle is the collection of points in a plane that are the same distance from a given point in the plane." is true.
