Question

A circle is the collection of points in a plane that are the same distance from a given point in the plane. true or false

Answer 1

Answer: TRUE.

Explanation:

It knows that the distance "d" between any two points is equal to
`√((x-x_0)^2-(y-y_0)^2)`.

In a circle the distance between a point that belongs to the circumference and the center (
`x_0,y_0`) is always the same. Then:

`√((x-x_0)^2-(y-y_0)^2)` is always equal to a constant called "r".

This is:

`√((x-x_0)^2-(y-y_0)^2)=r`

Square both sides of the equality:

`(√((x-x_0)^2-(y-y_0)^2))^2=r^2`

`(x-x_0)^2-(y-y_0)^2=r^2`

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.

Answer 2

true

any given point in a circle is equal distance from the center